Cooperative Threads with Effective-Address in Simulated ...mcruz/thread.pdf · 7/29/2019 · communication using MPI. In [20], the researchers present a genetic hybrid island model

applied sciences

Article

Cooperative Threads with Effective-Address inSimulated Annealing Algorithm to Job ShopScheduling Problems

Marco Antonio Cruz-Chávez 1,* , Jesús del C. Peralta-Abarca 2 and Martín H. Cruz-Rosales 3

1 Research Center in Engineering and Applied Sciences, Autonomous University of Morelos State (UAEM),Avenida Universidad 1001 Colonia Chamilpa, C.P. Cuernavaca 62209, Morelos, Mexico

2 Faculty of Chemical Sciences and Engineering, UAEM, Avenida Universidad 1001 Colonia Chamilpa, C.P.Cuernavaca 62209, Morelos, Mexico

3 Faculty of Accounting, Administration & Informatics, UAEM, Avenida Universidad 1001 Colonia Chamilpa,C.P. Cuernavaca 62209, Morelos, Mexico

* Correspondence: [email protected]

Received: 14 July 2019; Accepted: 9 August 2019; Published: 15 August 2019��

Featured Application: This research allows improving the schedule of jobs in manufacturingworkshops and this increases the amount of products without having the need to increase theproduction machines.

Abstract: This paper presents a parallel algorithm applied to the job shop scheduling problem(JSSP). The algorithm generates a set of threads, which work in parallel. Each generated thread,executes a procedure of simulated annealing which obtains one solution for the problem. Eachsolution is directed towards the best solution found by the system at the present, through a procedurecalled effective-address. The cooperative algorithm evaluates the makespan for various benchmarksof different sizes, small, medium, and large. A statistical analysis of the results of the algorithmis presented and a comparison of performance with other (sequential, parallel, and distributedprocessing) algorithms that are found in the literature is presented. The obtained results show thatthe cooperation of threads carried out by means of effective-address procedure permits to simulatedannealing to work with increased efficacy and efficiency for problems of JSSP.

Keywords: effective-address; hamming distance; thread; critical section

1. Introduction

The job shop scheduling problem (JSSP) appears in the manufacturing industry and is classified bythe theory of complexity as one of the most difficult problems to solve within the class of NP-completeproblems [1], where NP, indicates non-polynomial behavior problems, that is, problems that do nothave an exact algorithm that can solve them in polynomial time. Due to the difficulty of solvingthis problem, it is important to look for new alternatives in the solution thread in order to improvethe cost/performance of the existing algorithms that attempt this type of problem for the purpose ofobtaining better solutions.

In the solution of JSSP, very little information exists with regard to algorithms that apply parallelthreading as a form of cooperation in order to accelerate the search for better solutions. In [2], withtheir branch and bound algorithm, they generate a search tree through the cooperation of 16 threads,where each thread is executed by a CPU (central processing unit). In [3], the researchers investigatethe performance with multi-threading using mixed integer programming and parameter tuning withCPLEX to JSSP of 15 × 15, where CPLEX is a Simplex algorithm written in language C (C-Simplex

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resulted in CPLEX). In [4] with their branch and bound algorithm, the researchers propose a GPU-basedparallelization in which a two level scheme is used, therefore, at each iteration, several search treenodes are evaluated on the GPU using several thread-blocks to the blocking job shop schedulingproblem. In [5], the researchers present a parallel branch and bound which is based on implicitenumeration and parallel particle swarm optimization (PSO). In parallel PSO, the model used isthe master-slave model; each client evaluates its own population at the same time. The master isresponsible for defining the parameters that are necessary for the population in each slave. In eachiteration, each slave sends its best particle back to the master. The master sends the best receivedparticles to all the slaves to inform them of the best particle in the group. Then each slave updates itsbest particle. The parallel branch and bound use the logical ring topology. It involves a simulatedannealing algorithm to improve the upper bound when a timeout controller is used. The resultspresented for the parallel branch and bound are for small problems of up to 6 jobs and 15 machines.They only perform comparative tests of efficiency between the proposed sequential and parallel branchand bound. The results presented for the PSO are not good, and are only for medium sized problems.In [6], the researchers use a system of distributed computation compound for 12 personal computers toexecute an algorithm of simulated annealing (SA) and their results are not good. In [7], the researchersreport a parallel version of the simulated annealing algorithm using a procedure for parallelizing themost difficult calculation. In this case the most difficult calculation is to obtain the critical path inthe disjunctive graph model. In [8], one parallel algorithm of simulated annealing is reported, whichuses a distribution system of 20 computers; this algorithm manages the concept of populations ingenetic algorithms, but without using a genetics operator. Very few investigations of the application ofthe cooperation in algorithms for JSSP exist. In [9], a novel concept of island model with islands ofdifferent sizes is proposed to scheduling problems. The proposed multi-size approach to island modelmakes the effectiveness of the island-based algorithm independent from the particular population sizewhich is identical for all islands in the canonical island model, also eliminates the need to tune thepopulation size of identical islands to ensure high efficiency. In some genetic algorithms, cooperationis applied by means of crossover of individuals that belongs to a population. This is done withinlocal search procedures using deterministic algorithms (critical path algorithms) in order to evaluatethe neighborhood function. Satisfactory results can be obtained [10,11]. Some of the algorithms thathave applied cooperation with crossover in problems of JSSP have not given good results [11–13], thisis mainly due to the fact that the methods of crossover used, are based on improving the solutionwith the idea of crossed neighbors in order to get better neighbors when comparing neighbors withthe neighborhoods of origin, but not with the best neighbor available in the neighborhoods. Thefact of not considering the best solution available for the crossover causes, that with the passage oftime, the search is not directed toward a space of good solutions. Consequently, the final solution ofthe algorithm with cooperation generally is not satisfactory. In [14], the researchers present a hybridgenetic algorithm (GA) with parallel implementation of simulated annealing (SA) for the JSSP. Thealgorithm uses a GA to generate an initial population of solutions within a server machine. Everysolution in the population is then distributed to a client machine. The client machine runs an SA andsends the best obtained solution to the server machine. The server again uses the GA to improvethe population. This procedure GA-SA is repeated for a number of iterations. The presented resultsare for medium-sized benchmarks with relative errors below 3.5%. In [15], the researchers present ahybrid parallel genetic algorithm for the JSSP. The method deals with concepts of solution backbonesin order to intensify the search in promising regions. They utilize path-relinking and taboo searchin crossover and mutation operators. They also use the island model to store a population on eachisland. If after several iterations of the GA the fitness does not improve, then the subpopulationsstart to merge, always maintaining the size of the initial population, until in the end there is only onepopulation. In [16], the researchers present a hybrid parallel genetic micro algorithm. The parallelimplementation considers M independent GAs running M subpopulations with independent memories,genetic operations, and function evaluation (island models) in each generation. The best individuals

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in each subpopulation are broadcast to the other subpopulations over a communication network.Each subpopulation consists of small populations, which communicate among themselves and shareinformation through asynchronous message passing. Although a study is presented on the efficiency(speedup) of the parallel algorithm, it appears incomplete. The study evaluates the efficiency of theparallel algorithm for each increase in the number of threads, but the efficiency is averaged and istherefore not sufficient to evaluate the algorithm. In [17], the researchers present a parallel GA for theJSSP in which active schedules are created by the priority rules of Giffler and Thompson. The modelused is the master–slave model. The GA works iteratively with distributed slave populations. It doesnot apply cooperation between threads and sends the best results only to the master using messagepassing interface (MPI). The obtained results are good, and the benchmarks are of small and mediumsize. The cooperation with agents in SA for JSSP dynamics has also been applied, but the resultsin [18], show that the time consumed in communication between agents is extensive. If the existenceof several agents is considered, SA works more slowly than normal when the size of the problem ofJSSP is medium or bigger. Another application with agents is an agent-based parallel approach with agenetic algorithm. For this approach, the results for benchmarks of JSSP, which are small and medium,are very poor [19]. In this study, a layered architecture was used. In the lowest layer, a multi-agentsystem using JADE middleware was implemented which facilitated the creation of agents and theircommunication using MPI. In [20], the researchers present a genetic hybrid island model algorithm forJSSP, proposing a strategy based on the self-adaptation phase that generates a better balance betweendiversification and intensification. Such algorithm works through pseudo-parallel on a single processorsystem. In [21], the researchers present an effective island model genetic algorithm for JSSP, proposinga mechanism based on natural migratory selection that improves the search and avoids prematureconvergence. This algorithm works through pseudo-parallel on a single processor system. In [22],the researchers present an effective genetic algorithm with a critical-path-guided crossover operatorfor JSSP; they use the critical path in the global search method during the crossover operator. In [23],the researchers present a hybrid simulated annealing for the job shop scheduling problem. It is basedon an ant colony that generates the initial population and a simulated annealing that improves thepopulation. The algorithm works in a parallel computer with a genetic algorithm, using a crossovergenetic operator inside the cycle of temperature in simulated annealing. In [24], the researchers presenta parallel artificial bee colony algorithm for JSSP with a dynamic migration strategy. It determineswhen a colony should communicate with its neighboring colonies. Such algorithm is carried out invarious colonies in parallel. In [25], the researchers present a hybrid micro-genetic multi-populationalgorithm with collective communication that uses a set of elite micro-populations with computationalprocesses and improves individuals with simulated annealing.

In this paper, cooperation is applied in order to direct the search for a solution toward a spacewhere there are good solutions to the problem. A set of simulated annealing (SSA) is generated withrestart [26], where every time that a simulated annealing (SA) ends, an effective-address procedurewith the solution Si obtained from that SA and the best solution Sbest obtained by the SSA up to thatmoment is applied. The search will be directed toward a space where there are good solutions, makingthe new solution obtained by the effective-address procedure more similar to the best solution obtainedby the SSA.

Section one gives a brief introduction. Section two describes the formulation of the job shopscheduling problem. Section three presents the proposed simulated annealing algorithm that appliescooperative threads (SACT) and the effective-address procedure, where a CREW model (concurrent read,exclusive write) is used in implementation [27]. Section four shows the experimental results. Finally,the conclusions drawn by the present work are explained.

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2. Disjunctive Formulation of Job Shop Scheduling Problem

The disjunctive formulation has several sets: A set J of n jobs, where, J = {J1, J2, . . . , Jn}; a set M ofm machines where M = {M1, M2, . . . , Mm}; and a set O of operations where O = {1, 2, 3, . . .}. Theseoperations form k subsets of operations for each one of the jobs (Jk ⊆ O) and machines (Mk ⊆ O).

Each operation i has a processing time of pi. In a job Jk, each pair of operations i, j possesses arelationship of precedence (i < j). Only one operation performed by a machine Mk, can be executedat any given point in time. Given the previously mentioned problem restrictions, the function of thestarting time, s of each operation can be represented in the following manner:

The constraint in (1) indicates that the starting time of the operation j must be greater than or equalto zero; meaning only positive values are accepted. The constraint in (2) is a precedence constraint.It indicates that within one job which contains operations i and j, in order for j to begin, i must becompleted. The constraints in (3) are disjunctive. These constraints ensure that two operations, i and j,which are performed by the same machine Mk are not carried out simultaneously. The constraints in (4)indicate that the time necessary to complete an operation j must be less than or equal to the makespan(Cmax). The objective is to minimize the makespan, defined as the maximum time of completion of thelast job, which is determined according to the starting times, and can be expressed as (5). The timeunits in the makespan are open, thus they can be assigned as seconds, minutes, hours, and so on, aslong as these are consistent units. In this work the makespan is simply labeled as units of time.

∀i ∈ O, si ≥ 0 (1)

∀(i, j) ∈ O∧ (i ≺ j) ∈ Jk (si + pi) ≤ s j (2)

∀(i, j) ∈ O ∧ (i, j) ∈Mk (si + pi) ≤ s j ∨(s j + p j

)≤ si (3)

∀i ∈ O (si + pi) ≤ Cmax (4)

min( f ) = min[ maxi ∈ O

(si + pi)]

(5)

Figure 1 shows the disjunctive graph model G = (A, E, O) for a JSSP of 3 × 3 (three machines andthree jobs). This disjunctive graph is formed by three sets. The operations set, O, is made up of thenodes in G, numbered one to nine. The processing time appears next to each operation. The beginningand ending operations (I and * respectively) are fictitious, with processing times equal to zero. The setA is composed of conjunctive arcs, each one of these arcs unites a pair of operations that belongs to thesame job Ji and each one represents a precedence constraint. The operations 1, 2, and 3 are connectedby these arcs and therefore form job one (J1). Jobs two (J2), and three (J3) are connected in the sameway with operations (4, 5, 6) and (7, 8, 9), respectively. Each clique that belongs to E contains a subsetof operations that are executed by the same machine (Mk). Operations 1, 6, and 7 are executed by M1and generate a clique. Machines two (M2) and three (M3) generate a clique, each one with operations(2, 5, 8) and (3, 4, 9), respectively. The resources capacity constraints are represented between each pairof operations in E.

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Figure 1. Job shop scheduling with three jobs and three machines.

Figure 1. Job shop scheduling with three jobs and three machines.

3. Simulated Annealing Algorithm with Cooperative Threads (SACT)

The SACT algorithm (simulated annealing with cooperative threads) generates threads (H1, . . . ,Hn) that are administered by the operating system Windows Vista Ultimate 64 bits and which workin parallel form. Each thread generated Hi by SACT executes and controls one simulated annealingalgorithm (SA) in individual form. A core of CPU carries out the control of each thread Hi. If there aremore threads than cores of CPU’s, then the threads are distributed in a balanced way among the groupof cores of CPU’s that the computer contains. The cooperation among the existing threads, n, allowseach thread, Hi, in which simulated annealing is carried out, to know (in any instant of important time)the best solution, Sbest that has been found up to that point in time. This makes it possible that a threadHi can use the best solution Sbest to realize an effective-address procedure with the solution SHi_c that isfound by executing the SA algorithm. The resulting solution of the effective-address procedure is usedas an initial solution in the restart of SA within the thread Hi. Because SACT constantly needs to knowand upgrade the best solution Sbest. The threads have the advantage of working actively and using thesame memory address space. This makes the communication among them quicker [28]. The threadsare able to work simultaneously.

In SACT, the threads have continuous access to a space of shared memory that stores the bestfound solution for the n threads at a time t. This makes it necessary to implement a CREW model,where the access to this space of shared memory must be synchronized between the threads in order toavoid conflicts of reading/writing that could cause errors or loss of information. In order to synchronizethe access to this memory, the critical section is used. This guarantees that in any given instant in time,only one thread can have access to read/write, leaving the others on a waiting list.

In the operating system Windows Vista Ultimate 64 bits, the threads are administrated like objects,which are created for calling functions to the applications programming interface (API), using the MFClibrary [29]. In the SACT algorithm, the functions of the MFC of Microsoft visual C++, 2017, are usedfor the creation of the threads in a single process.

Each thread created by SACT works by carrying out the same number of identical activities, butwith different data. Knowing this, a class was created that upon generating n instances for the class, nobjects are generated. Each object represents a thread [30].

SACT has the possibility of generating a parent thread, which in turn generates n offspring threads,executing them in parallel form.

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The SACT algorithm applies to JSSP when the objective function is minimizing the maximumtime of completion of the last job, this is known as minimizing the makespan. Algorithm 1, presentsthe SA algorithm that is executed for each created thread Hi by SACT.

Algorithm 1 Simulated annealing executed in each generated thread Hi by SACT.

1. Obtain SHi, where SHi = f (UB).2. Give an iteration initial iter = 0, initialize values of Cf, Co, α, f(SHi_c).3. While (iter < Maxiter)

3.1. iter = iter+13.2. C = Co

3.3. While (C < Cf).

3.3.1. While (equilibrium does not exist)

3.3.1.1. S’Hi = permutation(SHi)

3.3.1.2. If(f(S’Hi) − f(SHi) ≤ 0) SHi = S’

Hi

3.3.1.3. If(f(S’Hi) − f(SHi) > 0) the state is accepted with the probability

Paccept = e−(f (S′Hi)− f (SHi)

C )

3.3.1.4. ρ = random number between (0, 1)3.3.1.5. If (ρ < Paccept) SHi = S’

Hi

3.3.1.6. If (f(SHi) < f(SHi_c)) SHi_c = SHi

3.3.1.7. If (f(SHi_c) < f(Sbest)) Sbest = SHi_c

3.3.2. C = C*α.

4. SHi = effective-address (SHi_c, Sbest).

The explanation of the algorithm is the following (Algorithm 1):

1. SHi is the schedule (initial solution of JSSP) obtained by the scheduling algorithm [12] using thedisjunctive graph model. The makespan of SHi should not surpass an upper bound (UB) definedas date of entrance. In order to define this bound see [26].

2. iter (global variable known in the n threads) is the variable that counts the number of SA executedin SACT. The final control coefficient, Cf, the initial control coefficient (temperature), Co, thecontrol factor α and the best f(SHi_c) (at the beginning this is a very large value) are initialized.

3. If the number of SA (iter) for SACT has not been completed (which is indicated with Maxiter),then a new SA is begun using the new solution SHi obtained with the effective-address procedure.

3.1. Begin the annealing iter.3.2. The value of initial control coefficient of the SA is initialized.3.3. The external cycle begins, which carries out the decrease in control coefficient (3.3.2) of the

SA according to α.

3.3.1. The internal cycle in annealing begins, which executes the Metropolis algorithmuntil equilibrium is reached, this depends on the size of the Markov chain (MC)and that for optimization problems is represented by the neighborhood size of asolution of the problem.

3.3.1.1.A neighborhood structure is used (explained later on, Section 3.1). Thisgenerates a state in annealing (neighbor S’Hi).

3.3.1.2.S’Hi is accepted as a new configuration if the energy of the system decreases.

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3.3.1.3.If the energy of the system increases, S’Hi is accepted as a new configurationaccording to the probability of acceptance Paccept obtained by the function ofBoltzmann, and

3.3.1.4. Comparing this Paccept with ρ, which is a random number uniformlydistributed between (0, 1).

3.3.1.5.If ρ < Paccept the generated state is accepted as the current state.3.3.1.6.If the new schedule cost f(SHi) is better than the best schedule cost f(SHi_c)

of the thread Hi, then SHi_c is upgraded.3.3.1.7.If the new schedule cost f(SHi) is better that the best schedule cost f(Sbest)

which has been obtained from all the threads of SACT, then Sbest is upgraded.

3.3.2. The control coefficient is decreased.

3.4. Every time that the thread Hi in execution finishes an SA, an effective-address is carried outbetween the best solution SHi_c obtained from the SA and the best existing solution Sbest inthe algorithm SACT. The effective-address mechanism is explained later (Section 3.2).

The final solution of the algorithm SACT is Sbest, which is obtained by the cooperation of the nthreads generated with SACT.

3.1. Neighborhood Generation Mechanism

Figure 2 presents the way to generate new neighbors S’ starting from a schedule S. From S, amachine Mk is chosen randomly and a pair of adjacent operations i, j is also selected randomly. Thepair is checked to make sure that slack time between i, j does not exist or fictitious slack time exit [30].This pair of operations is perturbed by exchanging their order of precedence and then the resultingschedule is checked for feasibility. If the schedule is feasible, it is considered to be a neighbor S’. If theschedule is not feasible, slack time or fictitious slack time between i, j exist. In this case, the thread isrepeated with a different pair of adjacent operations in a randomly chosen machine until a feasibleneighbor S’ is obtained.

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is repeated with a different pair of adjacent operations in a randomly chosen machine until a feasible neighbor S’ is obtained.

Figure 2. Generation of neighbors.

3.2. Effective-Address Mechanism

The thread Hi carries out the effective-address procedure every time that it finishes the execution of a SA. This effective-address procedure is carried out between the solution SHi_c of Hi, obtained by the SA, and the best solution, Sbest, found by the n threads. The effective-address procedure is the following:

The neighborhood structure N1 [31] is used in order to generate all the neighbors S’best of Sbest. These neighbors are always feasible because they only involve exchanging a pair of operations that belong to the critical path [31]. The similarity between each pair of solutions (schedules) SHi_c and a neighbor S’best is measured based on the Hamming distance. This distance is the number of arcs (in the digraph for the solution that results from Figure 1) that have different addresses between SHi_c and its neighbor S’best. In other words, it is the number of differences in the order of execution of operations on each machine [12]. There is a high probability of choosing an S’best of Sbest that has great similarity to SHi_c. This probability is generated in the following way. Three lists are created (L1, L2, L3) that store the S’best neighbors (generated with Sbest) according to the degree of similarity with SHi_c. The size of each list is different, L1 is the smallest, and thus the most similar to SHi_c whereas L3 has the neighbors which are less similar to SHi_c. L1 is chosen with more frequency in order to direct the search toward a good solution space. The size of each list is assigned according to the problem to be solved and it is defined experimentally (trial and error). In order to select one of the 3 lists, a number α at random with a value between 1 and 10 is generated. L1 is chosen if α is between 1 and α1. L2 is chosen if α is between α1 + 1 and α2. L3 is chosen if α is between α2 + 1 and 10. The size of each interval is |1 to α1| > | α1 + 1 to α2| > | α2 + 1 to 10|. Once the list is chosen, the neighbor S’ is chosen randomly from the list. This is the solution that is used in the beginning of the new SA for the thread Hi. The neighbor selected, S’best, is the result of the effective-address procedure carried out implicitly between SHi_c and Sbest. Naturally, this effective-address procedure is applied in each one of the n threads.

Figure 3 presents the running threads in SACT. First, a symbolic representation (SR) of the JSSP is generated. Then, a random solution SHi of SR is generated for each Hi thread. At the beginning of the algorithm, the best global solution Sbest, has a very large initial value defined with a lower bound (LB), which is not less than the sum of the processing time of all operations that are executed on the JSSP.

Figure 2. Generation of neighbors.

3.2. Effective-Address Mechanism

The thread Hi carries out the effective-address procedure every time that it finishes the execution ofa SA. This effective-address procedure is carried out between the solution SHi_c of Hi, obtained by the SA,and the best solution, Sbest, found by the n threads. The effective-address procedure is the following:

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The neighborhood structure N1 [31] is used in order to generate all the neighbors S’best of Sbest.These neighbors are always feasible because they only involve exchanging a pair of operations thatbelong to the critical path [31]. The similarity between each pair of solutions (schedules) SHi_c and aneighbor S’best is measured based on the Hamming distance. This distance is the number of arcs (in thedigraph for the solution that results from Figure 1) that have different addresses between SHi_c and itsneighbor S’best. In other words, it is the number of differences in the order of execution of operationson each machine [12]. There is a high probability of choosing an S’best of Sbest that has great similarityto SHi_c. This probability is generated in the following way. Three lists are created (L1, L2, L3) that storethe S’best neighbors (generated with Sbest) according to the degree of similarity with SHi_c. The size ofeach list is different, L1 is the smallest, and thus the most similar to SHi_c whereas L3 has the neighborswhich are less similar to SHi_c. L1 is chosen with more frequency in order to direct the search toward agood solution space. The size of each list is assigned according to the problem to be solved and it isdefined experimentally (trial and error). In order to select one of the 3 lists, a number α at randomwith a value between 1 and 10 is generated. L1 is chosen if α is between 1 and α1. L2 is chosen if α isbetween α1 + 1 and α2. L3 is chosen if α is between α2 + 1 and 10. The size of each interval is |1 to α1| >

| α1 + 1 to α2| > | α2 + 1 to 10|. Once the list is chosen, the neighbor S’ is chosen randomly from thelist. This is the solution that is used in the beginning of the new SA for the thread Hi. The neighborselected, S’best, is the result of the effective-address procedure carried out implicitly between SHi_c andSbest. Naturally, this effective-address procedure is applied in each one of the n threads.

Figure 3 presents the running threads in SACT. First, a symbolic representation (SR) of the JSSP isgenerated. Then, a random solution SHi of SR is generated for each Hi thread. At the beginning of thealgorithm, the best global solution Sbest, has a very large initial value defined with a lower bound (LB),which is not less than the sum of the processing time of all operations that are executed on the JSSP.Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 41

Figure 3. Communication of threads in SACT for sharing information using critical sections.

Each thread Hi, independently executes an SA with its own initial solution SHi. The optimized solution SHi_c of Hi, is obtained by the SA. Through critical sections, there is communication between threads to share and use the best global solution. There are two critical sections. In the first critical section, the information about the best solution for each thread (SHi_c) is selected, and from those best solutions (SHi_c vs. Sbest), one is chosen as the best global solution and is updated Sbest with the best solution. This best global solution (Sbest) is updated constantly, every time a thread enters the critical section with a better solution. The stopping criterion defines a maximum number of iterations with the Maxiter variable, which indicates the maximum number of SA runs performed by the total of the threads in the SACT algorithm. The sum of all iterations generated by all threads Hi, ... Hn, is defined with the IT variable, and this must not be greater than Maxiter. In the second critical section, when SACT continues execution to evaluate the effective-address in a thread, the best global solution used is the one being considered at that moment. The next time a thread evaluates the effective-address, it is possible that an improved global solution (Sbest) is used. This is because the execution of SACT is asynchronous. The advantage of SACT being asynchronous is that the throughput of the program is improved, which allows better exploration in the solution space because SACT is not concentrated only on a better solution (Sbest). When Sbest is improved, it is used for the following threads that require evaluation of the effective-address. This configuration improves the results obtained by SACT. The evaluation of the effective-address allows the realization of a search with SA(SHi) in a region of the solution space which contains good solutions for the problem to be resolved.

4. Experimental Results

For the experimental tests, we used a workstation PowerEdge T320, Intel® Xeon® Processor E5-2470 v2 (2200 Mission College Blvd. Santa Clara, CA 95054-1549, USA) with 10 cores of 3.10 GHz each, 24 GB RAM, 167 GB HD, Windows Vista Ultimate 64 bits, Visual C++ 2008, and MFC library to generate threads.

This type of system employs CISC architecture (Complex Instruction Set Computer) and is the one currently operating with high-powered applications, such as super-computing. This is unlike the ARM architecture (Advanced RISC Machines), which is enjoying great popularity and works better

Figure 3. Communication of threads in SACT for sharing information using critical sections.

Each thread Hi, independently executes an SA with its own initial solution SHi. The optimizedsolution SHi_c of Hi, is obtained by the SA. Through critical sections, there is communication betweenthreads to share and use the best global solution. There are two critical sections. In the first critical

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section, the information about the best solution for each thread (SHi_c) is selected, and from those bestsolutions (SHi_c vs. Sbest), one is chosen as the best global solution and is updated Sbest with the bestsolution. This best global solution (Sbest) is updated constantly, every time a thread enters the criticalsection with a better solution. The stopping criterion defines a maximum number of iterations withthe Maxiter variable, which indicates the maximum number of SA runs performed by the total of thethreads in the SACT algorithm. The sum of all iterations generated by all threads Hi, ... Hn, is definedwith the IT variable, and this must not be greater than Maxiter. In the second critical section, whenSACT continues execution to evaluate the effective-address in a thread, the best global solution usedis the one being considered at that moment. The next time a thread evaluates the effective-address, itis possible that an improved global solution (Sbest) is used. This is because the execution of SACT isasynchronous. The advantage of SACT being asynchronous is that the throughput of the program isimproved, which allows better exploration in the solution space because SACT is not concentratedonly on a better solution (Sbest). When Sbest is improved, it is used for the following threads that requireevaluation of the effective-address. This configuration improves the results obtained by SACT. Theevaluation of the effective-address allows the realization of a search with SA(SHi) in a region of thesolution space which contains good solutions for the problem to be resolved.

4. Experimental Results

For the experimental tests, we used a workstation PowerEdge T320, Intel® Xeon® ProcessorE5-2470 v2 (2200 Mission College Blvd. Santa Clara, CA 95054-1549, USA) with 10 cores of 3.10 GHzeach, 24 GB RAM, 167 GB HD, Windows Vista Ultimate 64 bits, Visual C++ 2008, and MFC library togenerate threads.

This type of system employs CISC architecture (Complex Instruction Set Computer) and is theone currently operating with high-powered applications, such as super-computing. This is unlikethe ARM architecture (Advanced RISC Machines), which is enjoying great popularity and worksbetter on battery-powered devices nowadays: On mobile phones and tablets, for instance. The ARMarchitecture has fewer transistors than CISC architecture processors and significantly reduces costs,heat, and energy during its operation. ARM architecture is expected to lead its way into creatingsuper-computing equipment given the great benefits it can bring within this area. Nvidia’s GPUs arecurrently used for CUDA-programmed supercomputing and Nvidia is starting to work on supportingARM processors to create new high performance computing computers that manage energy moreefficiently, as well as numerical processing from 2020 [32].

4.1. JSSP Instances and Simmulated Annealing Parameters

The instances of job shop used were 48 benchmarks of different sizes from the OR-library [33],nineteen small-sized instances, fifteen medium-sized instances, fourteen large-size instances.Additionally, ten large-size benchmarks of [34] were used, DMU-06-DMU10, and DMU46-DMU50.In this work only square instances are used, considering that square instances are more difficult tosolve [35]. Table A1, presented in Appendix A, shows benchmarks used to evaluate the proposedSACT algorithm. The names and optimal values or known upper bound of each instance are presented.

For each benchmark, SACT performed 30 executions, with Maxiter = 2500, or less if theoptimal/upper bound value is found and works with 48 threads.

SACT algorithm used SA parameters as C0, Cf, α, MC (Markov chain), and UB (solution at thebeginning, makespan), were obtained with a sensibility analysis, Table 1 shows the results for allinstances. In effective-address mechanism, the size of the three lists was the same for all the instances:For L1 it was 20%, for L2 it was 30%, and for L3 it was 50%, with regard to the total neighbors S’ in theneighborhood. α1 = 5, α2 = 8, and α3 = 10. The above are the conditions applied on each of the 30 testsperformed for each benchmark.

Appl. Sci. 2019, 9, 3360 10 of 40

Table 1. Tuned parameters of simulated annealing.

Problem Co Cf α MC UB = f(S)

Size 6 × 6

FT06 800 1.0 0.98 30 80

Size 10 × 10

FT10 25 1.0 0.98 1000 2500ORB and ABZ 64,000 1.0 0.98 1000 2500

Size 15 × 15

LA16 to LA20 64,000 1.0 0.98 1000 2500LA36 to LA40 2 1 × 10−6 0.99 300 2500TA01 to TA10 25 1.0 0.98 800 2500

Size 20 × 20

TA21 to TA30 2 1 × 10−6 0.99 300 3500YN 2 1 × 10−6 0.99 300 2000

DMU06 to DMU10 2 5 × 10−6 0.99 300 9000DMU46 to DMU50 100 0.05 0.99 6000 9500

4.2. Effect of Cooperating with Effective-Address

In this paper, two different types of cooperation are utilized. In the first type of cooperation, SACTuses an effective-address procedure between two solutions with the best global solution Sbest obtainedfrom all threads. The best solution in the running thread is SHi_c, i.e., SHi = effective-address (SHi_c, Sbest).With solution SHi obtained in each thread Hi, SA is restarted in each thread Hi. The second type ofcooperation does not use effective-address in the SACT procedure for restarting SA in each thread Hi.Instead, SHi = SHi_c is used. In addition, in each iteration of SACT, the best solution Sbest is saved.Cooperation occurs because the n threads working in parallel to find a solution at different points inthe solution space always keep the best solution found in Sbest.

In both types of cooperation, the best global solution Sbest is obtained for each iteration of SACTbecause the best solution found in all threads for each SA is compared with the current best globalsolution, and the best solution Sbest is saved.

The SACT algorithm was executed with several different numbers of threads and with a maximumvalue of 2500 Maxiter SA, which corresponds to about an hour of runtime for a number of threadsSA ≥ 8. Figures 4–12 (Maxiter vs. Sbest), present respectively the performance of the cooperation ofthreads in SA for the problems YN1, YN2, YN3, and YN4. These figures present the execution of SACTfor 8, 16, 32, 40, and 48 threads.

Figures 4a, 5a, 6a and 7a show the relationship with efficacy. When there is cooperation withoutusing the effective-address procedure, it can be observed that the best result obtained when Maxiterreaches 2500 SA does not depend on the number of threads in execution in SACT. The value of themakespan is very similar for any number of threads. This may be because there is no direct way toarrive at the same solution.

Figures 4b, 5b, 6b and 7b show the relationship with efficacy. It can be observed that the best resultobtained within the Maxiter = 2500 SA, using the effective-address procedure, was when SACT used48 threads. Figure 6b shows that in YN3, 48 threads compete with 40 threads to find the best solution.This indicates that with a greater number of threads, on average, better solutions may be obtained.This effect is seen because the larger group of threads (H48) has the same directed pathway toward thesame Sbest solution. This is due to the greater number of effective-address applied to the n threads withthe best solution Sbest. The result is that a greater number of threads lead to a directed path through thebest solution Sbest. The Hamming distance becomes smaller between the set of threads with respectto Sbest. This can be seen in Figures 8b, 9b, 10b and 11b. It is generally observed in all cases that onaverage, the Hamming distance decreases with an increasing number of threads. Interestingly, with

Appl. Sci. 2019, 9, 3360 11 of 40

regard to the efficacy of the algorithm that can be noted in Figures 4b and 7b for all other threads (H8,H16, H32, and H40), there is no defined behavior. For example, the efficacy as a function of the numberof threads for YN1 (Figure 4b) in increasing order is H40, H8, H16, H32, H48. Efficacy as a function ofthe number of threads for YN2 (Figure 5b) in increasing order is H8, H16, H32, H40, H48. Efficacy as afunction of the number of threads and YN3 (Figure 6b) in increasing order is H8, H16, H32, H48, H40.Efficacy as a function of the number of threads for YN4 (Figure 7b) in increasing order is H40, H16,H8, H32, H48. It can be seen that the efficacy of the algorithm SACT based on the number of threads,depends on the problem to solve. However, it can be observed that for all problems YN, generallywith 48 threads, the efficacy of SACT is better in almost all YN; the average efficacy is better using 48threads (H48).

Figures 4b, 5b, 6b and 7b show that the worst behavior occurs when SACT is executed in asingle thread sequence (H1) where cooperation with the effective-address procedure is with the onlythread that exists (H1). It is noted that approximately from iteration = 100 SA, the efficiency decreasesconsiderably for finding a good makespan when H8 is compared to H48. This behavior occurs forthe four problems YN. The opposite happens when there is cooperation thread (H8 to H48). Efficacyincreases on average from Figures 4b, 5b, 6b and 7b, by about 10 units from iteration 250 to 2500 SA.This makes these makespan results better than those found by H1. For example, in YN1 of Figure 4b,in iteration 500 SA, the best average makespan when there is cooperation (H32) is close to 898. Whenthere is no cooperation (H1), the makespan is about 909. In the iteration 2500 SA, the average makespanwhen there is cooperation (H48) is close to 894 and when there is no cooperation (H1), the makespan isclose to 906.

Figures 4a, 5a, 6a and 7a show that the worst performances are when SACT is executed in a singlethread sequence (H1), where cooperation does not use the effective-address procedure. When the onlythread that exists is (H1), the best solution is taken from the previous SA (SH1 = SH1_c). It is notedthat approximately at the beginning (before Maxiter = 50 SA), the efficacy of H1 is better. Then thisefficacy decreases considerably. This behavior occurs in all four YN problems. The opposite happenswhen there is cooperation thread (H8 to H48). The efficacy of cooperation increases as the number ofiterations increases of 250–500 SA, as shown by better results for the makespan as compared to thatfound by H1. For example, in YN1, Figure 4a, at iteration 500 SA, the average makespan when there iscooperation (H8) is close to 902 and when there is no cooperation (H1), the makespan is about 908. Initeration 2000 SA, the average makespan when there is cooperation (H40) is close to 896 and whenthere is no cooperation (H1), the makespan is near 906.

Appl. Sci. 2019, 9, 3360 12 of 40


makespan when there is cooperation (H8) is close to 902 and when there is no cooperation (H1), the makespan is about 908. In iteration 2000 SA, the average makespan when there is cooperation (H40) is close to 896 and when there is no cooperation (H1), the makespan is near 906.

(a)

(b)

Figure 4. YN1-20 × 20. Performance of cooperation with different numbers of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address. Figure 4. YN1-20 × 20. Performance of cooperation with different numbers of threads. Average of 30tests. (a) Without effective-address. (b) With effective-address.


(a)

(b) Figure 5. YN2-20 × 20. Performance of cooperation with different numbers of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address. Figure 5. YN2-20 × 20. Performance of cooperation with different numbers of threads. Average of 30tests. (a) Without effective-address. (b) With effective-address.


(a)

(b)

Figure 6. YN3-20 × 20. Performance of cooperation with different numbers of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address.4.

An interesting point related to the efficacy of the SACT algorithm is that when there is cooperation but no effective-address there is no defined behavior for different threads in execution, as seen in Figures 4a to 7a. Table 2 summarizes the efficacy of SACT with cooperation and without effective-address for YN problems in descending order. It can be observed that the efficacy of the algorithm SACT as a function of the number of threads depends on the problem being resolved. When there is both cooperation and effective-address, there is definite behavior for running threads, as can be seen in Figures 4b to 7b. Table 3 presents the efficacy of SACT with cooperation and effective-address for YN problems in descending order. It can be seen that the increased efficacy of the SACT algorithm, as a function of the number of threads, occurs when the number of threads is the largest (H48). The exception to this is the YN3 problem, in which case the number of threads is also great. In YN1 and YN4, the worst efficacy is obtained with H40. According to Table 4, in order to define the number of threads needed to increase the efficacy in SACT, it would be necessary to start with H32

Figure 6. YN3-20 × 20. Performance of cooperation with different numbers of threads. Average of 30tests. (a) Without effective-address. (b) With effective-address.4.

An interesting point related to the efficacy of the SACT algorithm is that when there is cooperationbut no effective-address there is no defined behavior for different threads in execution, as seen inFigures 4a, 5a, 6a and 7a. Table 2 summarizes the efficacy of SACT with cooperation and withouteffective-address for YN problems in descending order. It can be observed that the efficacy of thealgorithm SACT as a function of the number of threads depends on the problem being resolved. Whenthere is both cooperation and effective-address, there is definite behavior for running threads, as canbe seen in Figures 4b, 5b, 6b and 7b. Table 3 presents the efficacy of SACT with cooperation andeffective-address for YN problems in descending order. It can be seen that the increased efficacy of theSACT algorithm, as a function of the number of threads, occurs when the number of threads is thelargest (H48). The exception to this is the YN3 problem, in which case the number of threads is alsogreat. In YN1 and YN4, the worst efficacy is obtained with H40. According to Table 4, in order todefine the number of threads needed to increase the efficacy in SACT, it would be necessary to start

Appl. Sci. 2019, 9, 3360 15 of 40

with H32 threads, and slowly increase the threads, being careful detect cases when SACT has a numberof threads that presents a low efficacy, as in the case of YN1(H40) and YN4(H40).

An interesting point related to the efficacy of the SACT algorithm is that when there is cooperationbut no effective-address there is no defined behavior for different threads in execution, as seen inFigures 4a, 5a, 6a and 7a.

The efficacy of cooperation with the effective-address procedure (Figures 4b, 5b, 6b and 7b), isnoticed when, with a small number of iterations 500 SA, the value of an average makespan (MS)improves considerably when compared to the second type of cooperation (Figures 4a, 5a, 6a and 7a).For example, for YN1 with effective-address, the average MS is about 900 (H8, H16, H32, H40, H48), asseen in Figure 4b. For YN1 without effective-address, the average MS is close to 905 (H8, H16, H32, H40,H48) as shown in Figure 4a. For YN2 with effective-address, the average MS is close to 922 (H8, H16,H32, H40, H48), as noted in Figure 5b. For YN2 without effective-address, the average MS is close to 928(H8, H16, H32, H40, H48), as observed in Figure 5a.


threads, and slowly increase the threads, being careful detect cases when SACT has a number of threads that presents a low efficacy, as in the case of YN1(H40) and YN4(H40).

An interesting point related to the efficacy of the SACT algorithm is that when there is cooperation but no effective-address there is no defined behavior for different threads in execution, as seen in Figures 4a to 7a.

The efficacy of cooperation with the effective-address procedure (Figures 4b through Figure 7b), is noticed when, with a small number of iterations 500 SA, the value of an average makespan (MS) improves considerably when compared to the second type of cooperation (Figures 4a to 7a). For example, for YN1 with effective-address, the average MS is about 900 (H8, H16, H32, H40, H48), as seen in Figure 4b. For YN1 without effective-address, the average MS is close to 905 (H8, H16, H32, H40, H48) as shown in Figure 4a. For YN2 with effective-address, the average MS is close to 922 (H8, H16, H32, H40, H48), as noted in Figure 5b. For YN2 without effective-address, the average MS is close to 928 (H8, H16, H32, H40, H48), as observed in Figure 5a.

(a)

(b)

Figure 7. YN4-20 × 20. Performance of cooperation with threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address procedure.

Figure 7. YN4-20 × 20. Performance of cooperation with threads. Average of 30 tests. (a) Withouteffective-address. (b) With effective-address procedure.

Appl. Sci. 2019, 9, 3360 16 of 40

For YN3, with effective-address, the average MS is close to 906 (H8, H16, H32, H40, H48), as seen inFigure 6b. For YN3 without effective-address, the average MS is close to 912 (H8, H16, H32, H40, H48),as shown in Figure 6a. For YN4 with effective-address, the average MS is close to 922 (H8, H16, H32,H40, H48), as noted Figure 7b. For YN4 without effective-address, the average MS is close to 928 (H8,H16, H32, H40, H48), as seen in Figure 7a. It can be observed that with a Maxiter = 500, the increasedefficacy is almost constant for all YN, with 6 units of improvement in makespan when effective-addressis used, as compared to when it is not used.

When comparing the first type of cooperation (Figures 4b, 5b, 6b and 7b) and second type ofcooperation (Figures 4a, 5a, 6a and 7a), when the number of iterations is 2500, the average valueof MS is better for the first type of cooperation. For YN1 with effective-address, the average MS isapproximately 896 (H8, H16, H32, H40, H48), as seen in Figure 4b. For YN1 without effective-address,the average MS is approximately 898 (H8, H16, H32, H40, H48), as shown in Figure 4a. For YN2with effective-address, the average MS is approximately 904 (H8, H16, H32, H40, H48), as noted inFigure 5b. For YN2 without effective-address, the average MS is approximately 905 (H8, H16, H32, H40,H48), as observed in Figure 5a. For YN3 with effective-address, the average MS is approximately 904(H8, H16, H32, H40, H48), as seen in Figure 6b. For YN3 without effective-address, the average MS isapproximately 905 (H8, H16, H32, H40, H48), as shown in Figure 6a. For YN4 with effective-address,the average MS is approximately 980 (H8, H16, H32, H40, H48), as observed in Figure 7b. For YN4without effective-address, the average MS is approximately 985 (H8, H16, H32, H40, H48), as noted inFigure 7a. With a Maxiter = 2500, the increase in efficiency of the problem varies from 1 to 5 units ofimprovement in MS whether effective-address is used or not used. Even the slightest improvement inMS, which is 1, is very good, because what is sought with any optimization algorithm is to improvethe bounds already known for big problems. The improved efficacy makes SACT competitive with thebest algorithms reported in the literature, according to the results presented below.

Table 2 summarizes the efficacy of SACT with cooperation and without effective-address for YNproblems in descending order. It can be observed that the efficacy of the algorithm SACT as a functionof the number of threads depends on the problem being resolved. When there is both cooperationand effective-address, there is definite behavior for running threads, as can be seen in Figures 4b, 5b, 6band 7b.

Table 2. Efficacy of SACT with cooperation and without effective-address procedure as a function ofthe threads executed.

ProblemThreads

Best→Worst

YN1 H40–H16 H16 H48 H8 H32 H1YN2 H40–H32 H32 H8 H48 H16 H1YN3 H32 H48 H40 H16 H8 H1YN4 H16 H32 H40 H48 H8 H1

Table 3 summarizes the efficacy of SACT with cooperation and with effective-address for YNproblems in descending order. It can be seen that the increased efficacy of the SACT algorithm, asa function of the number of threads, occurs when the number of threads is the largest (H48). Theexception to this is the YN3 problem, in which case the number of threads is also great. In YN1 andYN4, the worst efficacy is obtained with H40. According to Table 3, in order to define the number ofthreads needed to increase the efficacy in SACT, it would be necessary to start with H32 threads, andslowly increase the threads, being careful to detect cases when SACT has a number of threads thatpresents a low efficacy, as in the case of YN1(H40) and YN4(H40).

Appl. Sci. 2019, 9, 3360 17 of 40

Table 3. Efficacy of SACT with cooperation and with effective-address procedure as a function of thethreads executed.

ProblemThreads

Best→Worst

YN1 H48 H32 H16 H8 H40 H1YN2 H48–H40 H40 H32 H16 H8 H1YN3 H40–H48 H48 H32 H16 H8 H1YN4 H48 H32 H8 H16 H40 H1

An evaluation of the Hamming distance presented in Figures 8–11 for the problems YN1, YN2,YN3, and YN4, respectively, with different numbers of threads, shows the degree of similarity in eachsolution with respect to the best solution in each iteration. The Hamming distance between SHi_c andSbest is obtained by comparing the sequence of operations on each machine of JSSP. If the Hammingvalue of a solution SHi_c is zero regarding Sbest, then the solution SHi_c is the same as Sbest and they havethe same configuration. Conversely, if SHi_c, has a very large Hamming value e.g., 400 (maximum valuefor a solution of the problem YN), then Sbest is totally different from SHi_c. The greater the Hammingdistance in a set of solutions S = {SH1_c, SH2_c, SH3_c, ..., SHn_c} with respect to an Sbest solution, thegreater the diversity in the set of solutions S, where n is the number of running threads.

Figures 8a, 9a, 10a and 11a, show that for cooperation without effective-address, more diversityexists than in the case of cooperation with effective-address. There is less diversity when movementfollows a directed path via Sbest, as seen in Figures 8b, 9b, 10b and 11b.

In Figures 8a, 9a, 10a and 11a, the Hamming distance without effective-address is within a rangeof 250 to 380. This is the case in almost all the execution interval of SACT, with different numbers ofthreads, and for different problems. It is true for YN1, YN2 between 180 and 325, YN3 between 190 and320, and YN4 between 200 and 325. Given that in this kind of cooperation there is no effective-address,the diversity behavior remains almost constant throughout the runtime of SACT, with the exception ofthe first 250 s. It is seen that H8 and H16 have less diversity than all the other YN problems. Sincethere is not a directed path through the best solution Sbest, the Hamming distance does not present asignificant decrease over time. The behavior does differ in relation to the number of threads, as can beobserved in Figures 8a, 9a, 10a and 11a. In Figure 8a for YN1, the Hamming distance is greatest forH40 and least for H8. The other Hamming distances have almost the same average behavior (H16, H32,and H48).

In Figures 8a, 9a, 10a and 11a, for YN2, YN3, and YN4, respectively, Hamming distance has thesame average performance for H16, H32, H40, and H48. For H8, the Hamming distance is smaller.The behavior of the Hamming distance based on the number of threads is due to each thread involvinga solution of SA. Because of this, if a larger number of solutions are compared to Sbest, there couldbe a wider diversity. This is not always true because Sbest has no relationship with most of thesolutions generated by the thread. An example of this can be seen in Figure 8a, Hamming (H48) <

Hamming (H40).In Figures 8b, 9b, 10b and 11b, the Hamming distance with effective-address at the beginning of

SACT has a diversity of Hamming values up to 350. This is then reduced considerably as the executionprogresses, and SACT has values in the range of 70 to 150. Because there is such cooperation witheffective-address, the diversity is reduced because the search in the solution directs itself toward Sbestsolutions. The behavior of the Hamming distance based on the number of threads is due to each threadinvolving a solution of SA. Because of this, if a larger number of solutions are compared to Sbest, therecould be a wider diversity. However, as the time of implementation of effective-address, is reduced,solutions are directed toward a single path. This could be debatable. It can be observed in Figures 8b,9b, 10b and 11b and Figures 4b, 5b, 6b and 7b, that applying cooperation threads with a directed pathtoward the best solution, a greater number of threads find better solutions. Since the greater diversity

Appl. Sci. 2019, 9, 3360 18 of 40

of the solutions is achieved with a greater number of threads, this avoids a hasty convergence prior tofinding good solutions.Appl. Sci. 2019, 9, x FOR PEER REVIEW 18 of 41

(a)

(b)

Figure 8. YN1-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address. Figure 8. YN1-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a)Without effective-address. (b) With effective-address.


(a)

(b)

Figure 9. YN2-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address. Figure 9. YN2-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a)Without effective-address. (b) With effective-address.


(a)

(b) Figure 10. YN3-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address. Figure 10. YN3-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a)Without effective-address. (b) With effective-address.


(a)

(b) Figure 11. YN4-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a) Without effective-address. (b) With effective-address.

Figure 12 presents YN1 with H32 threads, with a long execution time of close to 2 h. It can be seen that the effective-address procedure still does not make the Hamming distance converge in SACT. What can be seen is that if SACT is left running an extremely long time, the Hamming distance will tend to zero.

Figure 11. YN4-20 × 20. Solution diversity with different number of threads. Average of 30 tests. (a)Without effective-address. (b) With effective-address.

Figure 12 presents YN1 with H32 threads, with a long execution time of close to 2 h. It can be seenthat the effective-address procedure still does not make the Hamming distance converge in SACT. Whatcan be seen is that if SACT is left running an extremely long time, the Hamming distance will tendto zero.

Appl. Sci. 2019, 9, 3360 22 of 40


Figure 12. Effect of the effective-address procedure on the convergence of SACT for long times. YN1-20 × 20. H32.

The obtained results to YN problems indicate that in order for SACT to work with higher efficiency, it requires a computer that has a greater number of threads in order to allow SACT to make a greater number of effective-address with a greater number of threads, using a larger value of Maxiter.

4.3. SACT Statistic Review

The bar plot is used in this paper as a tool to assess the frequency obtained from the solutions in terms of quality, the unit is no dimensional. The x-axis in the bar plot represents the value of the objective function of JSSP, which is the MS obtained by SACT.

Figure 13 shows the bar plots for YN1, YN2, YN3, and YN4 problems. In the four bar plots, SACT presents frequencies whose distribution is skewed to the left with respect to the midpoint.

(a)

(b)

0

2

4

6

8

10

12

14

889.0 894.0 899.0 904.0 909.0

Freq

uency

Makespan (units of time)

0

2

4

6

8

10

12

14

910.8 916.6 922.4 928.2 934.0

Freq

uency


Figure 12. Effect of the effective-address procedure on the convergence of SACT for long times. YN1-20× 20. H32.

The obtained results to YN problems indicate that in order for SACT to work with higher efficiency,it requires a computer that has a greater number of threads in order to allow SACT to make a greaternumber of effective-address with a greater number of threads, using a larger value of Maxiter.


The bar plot is used in this paper as a tool to assess the frequency obtained from the solutionsin terms of quality, the unit is no dimensional. The x-axis in the bar plot represents the value of theobjective function of JSSP, which is the MS obtained by SACT.

Figure 13 shows the bar plots for YN1, YN2, YN3, and YN4 problems. In the four bar plots, SACTpresents frequencies whose distribution is skewed to the left with respect to the midpoint.


Figure 12. Effect of the effective-address procedure on the convergence of SACT for long times. YN1-20 × 20. H32.

The obtained results to YN problems indicate that in order for SACT to work with higher efficiency, it requires a computer that has a greater number of threads in order to allow SACT to make a greater number of effective-address with a greater number of threads, using a larger value of Maxiter.


The bar plot is used in this paper as a tool to assess the frequency obtained from the solutions in terms of quality, the unit is no dimensional. The x-axis in the bar plot represents the value of the objective function of JSSP, which is the MS obtained by SACT.

Figure 13 shows the bar plots for YN1, YN2, YN3, and YN4 problems. In the four bar plots, SACT presents frequencies whose distribution is skewed to the left with respect to the midpoint.

(a)

(b)

0

2

4

6

8

10

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889.0 894.0 899.0 904.0 909.0

Freq

uency


0

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14

910.8 916.6 922.4 928.2 934.0

Freq

uency


Figure 13. Cont.


Figure 13. SACT, bar plots YN1, YN1, YN3, YN4. (a) SACT, YN1 (30 tests). (b) SACT YN2 (30 tests). (c) SACT, YN3 (30 tests). (d) SACT, YN4 (30 tests).

The frequency distribution of YN1 and YN3 (Figure 13a,c) is very similar to a normal distribution. The frequency distribution of YN2 (Figure 13b) and YN4 (Figure 13d) is a little skewed to the right with respect to the midpoint (arithmetic mean), this is important because it indicates that most results obtained are to the left of the midpoint value, so the makespan found in most of the tests is of good quality. For YN1, 12 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 889–894. For YN2, 16 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 911–917. For YN3, 14 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 897–903. For YN4, 15 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 973–978. There is data concentration near the arithmetic mean for the four benchmarks, YN1 to YN4.

The frequency distribution of DMU06, DMU07, DMU08, and DMU10 (Figure 14) is skewed to the right with respect to the midpoint (arithmetic mean). DMU06 (Figure 14a) shows a bimodal frequency distribution, the highest frequency is 8 and is at the midpoint with 3319.2. At the left of the bar plot the frequency is 2 and the quality results is 3276. DMU07 (Figure 14b) and DMU08 (Figure 14c) present a data distribution further from the arithmetic mean, as opposed to DMU06, DMU09 (Figure 14d), and DMU10 (Figure 14e). For DMU07, 12 out of 30 results have a makepan value better or equal to the midpoint. At the far left of the bar plot the frequency is 6 and the quality results is 3092. At the far right of the bar plot the frequency is 2 and the quality results is 3227. DMU06 and DMU10 have a higher dispersion in the data distribution. For DMU08, 8 results of 30 obtained, have a makespan value better or equal to the midpoint and this value is 3225. At the far left of the bar plot the frequency is 8 and the quality results is 3225. At the far right of the bar plot the frequency is 1 and the quality results is 3390. For DMU09, 8 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 3140–3159. At the far left of the bar plot the frequency is 5 and the quality results is 3140. At the far right of the bar plot the frequency is 3 and the quality results is 3235. For DMU10, 10 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 3015–3029. At the far left of the bar plot the frequency is 3 and the quality results is 3015. At the far right of the bar plot the frequency is 4 and the quality results is 3085.

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Figure 13. SACT, bar plots YN1, YN1, YN3, YN4. (a) SACT, YN1 (30 tests). (b) SACT YN2 (30 tests). (c)SACT, YN3 (30 tests). (d) SACT, YN4 (30 tests).

The frequency distribution of YN1 and YN3 (Figure 13a,c) is very similar to a normal distribution.The frequency distribution of YN2 (Figure 13b) and YN4 (Figure 13d) is a little skewed to the rightwith respect to the midpoint (arithmetic mean), this is important because it indicates that most resultsobtained are to the left of the midpoint value, so the makespan found in most of the tests is of goodquality. For YN1, 12 results out of 30 obtained, have a makespan value better or equal to the midpointand are within the range 889–894. For YN2, 16 results out of 30 obtained, have a makespan value betteror equal to the midpoint and are within the range 911–917. For YN3, 14 results out of 30 obtained, havea makespan value better or equal to the midpoint and are within the range 897–903. For YN4, 15 resultsout of 30 obtained, have a makespan value better or equal to the midpoint and are within the range973–978. There is data concentration near the arithmetic mean for the four benchmarks, YN1 to YN4.

The frequency distribution of DMU06, DMU07, DMU08, and DMU10 (Figure 14) is skewed to theright with respect to the midpoint (arithmetic mean). DMU06 (Figure 14a) shows a bimodal frequencydistribution, the highest frequency is 8 and is at the midpoint with 3319.2. At the left of the bar plot thefrequency is 2 and the quality results is 3276. DMU07 (Figure 14b) and DMU08 (Figure 14c) presenta data distribution further from the arithmetic mean, as opposed to DMU06, DMU09 (Figure 14d),and DMU10 (Figure 14e). For DMU07, 12 out of 30 results have a makepan value better or equal tothe midpoint. At the far left of the bar plot the frequency is 6 and the quality results is 3092. At thefar right of the bar plot the frequency is 2 and the quality results is 3227. DMU06 and DMU10 havea higher dispersion in the data distribution. For DMU08, 8 results of 30 obtained, have a makespanvalue better or equal to the midpoint and this value is 3225. At the far left of the bar plot the frequencyis 8 and the quality results is 3225. At the far right of the bar plot the frequency is 1 and the qualityresults is 3390. For DMU09, 8 results out of 30 obtained, have a makespan value better or equal to themidpoint and are within the range 3140–3159. At the far left of the bar plot the frequency is 5 and thequality results is 3140. At the far right of the bar plot the frequency is 3 and the quality results is 3235.For DMU10, 10 results out of 30 obtained, have a makespan value better or equal to the midpoint andare within the range 3015–3029. At the far left of the bar plot the frequency is 3 and the quality resultsis 3015. At the far right of the bar plot the frequency is 4 and the quality results is 3085.


(a)

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Figure 14. SACT bar plots DMU06, DMU07, DMU08, DMU09, DMU10. (a) SACT, DMU06 (30 tests). (b) SACT, DMU07 (30 tests). (c) SACT, DMU08 (30 tests). (d) SACT, DMU09 (30 tests). (e) SACT, DMU10 (30 tests).

The frequency distribution of DMU46 and DMU47 (Figure 15) is skewed to the left with respect to the midpoint (arithmetic mean). For DMU48, the data are concentrated near the arithmetic mean, 26 out of 30. DMU50 has a central frequency distribution, with the exception of a single data that is far from this distribution. For DMU46 (Figure 15a), only 6 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 4143–4163. At the far left of the bar plot the frequency is 2 and the quality results is 4143. At the far right of the bar plot the frequency is 7 and the quality results is 4193. For DMU47 (Figure 15b), only 6 results out of 30 obtained, have a

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Figure 14. SACT bar plots DMU06, DMU07, DMU08, DMU09, DMU10. (a) SACT, DMU06 (30 tests).(b) SACT, DMU07 (30 tests). (c) SACT, DMU08 (30 tests). (d) SACT, DMU09 (30 tests). (e) SACT,DMU10 (30 tests).

The frequency distribution of DMU46 and DMU47 (Figure 15) is skewed to the left with respect tothe midpoint (arithmetic mean). For DMU48, the data are concentrated near the arithmetic mean, 26out of 30. DMU50 has a central frequency distribution, with the exception of a single data that is farfrom this distribution. For DMU46 (Figure 15a), only 6 results out of 30 obtained, have a makespanvalue better or equal to the midpoint and are within the range 4143–4163. At the far left of the barplot the frequency is 2 and the quality results is 4143. At the far right of the bar plot the frequency is7 and the quality results is 4193. For DMU47 (Figure 15b), only 6 results out of 30 obtained, have amakespan value better or equal to the midpoint and are within the range 4036–4060. At the far left

Appl. Sci. 2019, 9, 3360 25 of 40

of the bar plot the frequency is 1 and the quality results is 4036. At the far right of the bar plot thefrequency is 4 and the quality results is 4096. For DMU48 (Figure 15c), 9 results out of 30 obtained,have a makespan value better or equal to the midpoint and are within the range 3878–3900. At the farleft of the bar plot the frequency is 1 and the quality results is 3878. At the far right of the bar plot thefrequency is 1 and the quality results is 3988. For DMU49 (Figure 15d), 12 results out of 30 obtained,have a makespan value better or equal to the midpoint and are within the range 3831–3849. At the farleft of the bar plot the best frequency is 1 and the quality results is 3831. At the far right of the bar plotthe frequency is 3 and the quality results is 3876. For DMU50 (Figure 15e), 15 results out of 30 obtained,have a makespan value better or equal to the midpoint and are within the range 3842–3868. At the farleft of the bar plot the best frequency is 1 and the quality results is 3842. At far right of the bar plot thefrequency is 6 and the quality results is 3907.


makespan value better or equal to the midpoint and are within the range 4036–4060. At the far left of the bar plot the frequency is 1 and the quality results is 4036. At the far right of the bar plot the frequency is 4 and the quality results is 4096. For DMU48 (Figure 15c), 9 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 3878–3900. At the far left of the bar plot the frequency is 1 and the quality results is 3878. At the far right of the bar plot the frequency is 1 and the quality results is 3988. For DMU49 (Figure 15d), 12 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 3831–3849. At the far left of the bar plot the best frequency is 1 and the quality results is 3831. At the far right of the bar plot the frequency is 3 and the quality results is 3876. For DMU50 (Figure 15e), 15 results out of 30 obtained, have a makespan value better or equal to the midpoint and are within the range 3842–3868. At the far left of the bar plot the best frequency is 1 and the quality results is 3842. At far right of the bar plot the frequency is 6 and the quality results is 3907.

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Figure 15. SACT bar plots DMU46, DMU47, DMU48, DMU49, DMU50. (a) SACT, DMU46 (30 tests). (b) SACT, DMU47 (30 tests). (c) SACT, DMU48 (30 tests). (d) SACT, DMU49 (30 tests). (e) SACT, DMU50 (30 tests).

The results obtained for the other small, medium, and large problems are presented in Table 4, Tables 5, and Table 6, respectively. The SACT used 48 threads. For each benchmark, 30 executions were carried out with SACT.

Table 4 presents the MS results with SACT for small benchmarks. For all the benchmarks the optimal solution value is reached (RE = 0). For most benchmarks the standard deviation of zero is obtained. The greatest standard deviation is obtained in ORB03 (σ = 6.9). For most benchmarks, the mode is the optimal solution obtained. The mode is the value that appears most often in the 30 tests. For most benchmarks, the median obtained is the optimal solution. If the median achieves the optimum, then at least half of the 30 tests achieved the optimal solution. The more difficult benchmarks that obtained the optimal solution according to the mode and median were ORB02 and ORB03. Table 4 shows, based on the standard deviation that the data distribution in each benchmark is completely homogeneous and the data are grouped very close to each other or with the same optimal value. For each benchmark in 30 tests, almost the same value for mean, median, and mode are found,, indicating a symmetrical distribution of data.

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Figure 15. SACT bar plots DMU46, DMU47, DMU48, DMU49, DMU50. (a) SACT, DMU46 (30 tests).(b) SACT, DMU47 (30 tests). (c) SACT, DMU48 (30 tests). (d) SACT, DMU49 (30 tests). (e) SACT,DMU50 (30 tests).

The results obtained for the other small, medium, and large problems are presented in Tables 4–6,respectively. The SACT used 48 threads. For each benchmark, 30 executions were carried outwith SACT.

Table 4 presents the MS results with SACT for small benchmarks. For all the benchmarks theoptimal solution value is reached (RE = 0). For most benchmarks the standard deviation of zero isobtained. The greatest standard deviation is obtained in ORB03 (σ = 6.9). For most benchmarks,the mode is the optimal solution obtained. The mode is the value that appears most often in the 30tests. For most benchmarks, the median obtained is the optimal solution. If the median achieves theoptimum, then at least half of the 30 tests achieved the optimal solution. The more difficult benchmarksthat obtained the optimal solution according to the mode and median were ORB02 and ORB03. Table 4shows, based on the standard deviation that the data distribution in each benchmark is completelyhomogeneous and the data are grouped very close to each other or with the same optimal value. Foreach benchmark in 30 tests, almost the same value for mean, median, and mode are found„ indicatinga symmetrical distribution of data.

Table 4. SACT results, small benchmarks JSSP (10 × 10).

Problem OptimumUnits of Time

BetterUnits of Time

WorseUnits of Time

MeanUnits of Time %RE σ t Sec Median Mode

FT6 55 55 55 55 0 0.0 0.03 55 55FT10 930 930 930 930 0 0.0 1.05 930 930LA16 945 945 945 945 0 0.0 5.1 945 945LA17 784 784 784 784 0 0.0 4.9 784 784L18 848 848 848 848 0 0.0 4.2 848 848

LA19 842 842 842 842 0 0.0 4.1 842 842LA20 902 902 902 902 0 0.0 4.34 902 902

ORB01 1059 1059 1059 1059 0 0.0 13.63 1059 1059ORB02 888 888 889 889 0 0.4 9720 889 889ORB03 1005 1005 1021 1017 0 6.9 1048 1020 1021ORB04 1005 1005 1005 1005 0 0.0 155 1005 1005ORB05 887 887 890 888 0 1.6 1354 887 887ORB06 1010 1010 1010 1010 0 0.0 59 1010 1010ORB07 397 397 397 397 0 0.0 4.95 397 397ORB08 899 899 899 899 0 0.0 10.33 899 899ORB09 934 934 934 934 0 0.0 5 934 934ORB10 944 944 944 944 0 0.0 4.84 944 944ABZ5 1234 1234 1234 1234 0 0.0 12.62 1234 1234ABZ6 943 943 943 943 0 0.0 4.23 943 943

Appl. Sci. 2019, 9, 3360 27 of 40

Table 5 presents the MS results with SACT for medium benchmarks. For all the benchmarks, theoptimum value is reached (RE = 0). For most benchmarks, a small value of standard deviation (σ < 7)is obtained. The exception is the LA38 problem with the highest standard deviation of 21; however,this value is less than half of the arithmetic mean (mean) and this represents little dispersion of thedata, however the dispersion of the data found by SACT in a frequency distribution for medium-sizedbenchmarks, is greater than the dispersion found for small-sized benchmark data. LA29 finds theoptimal solution in 54 s. LA40 finds in 37 min. TA07 finds the optimal solution in 99 s and TA07 findsin 85 min. More than half of the benchmarks of medium size have a mode equal to the optimal solution.For most benchmarks, the median obtained is not the optimal solution. The most difficult problemsfor obtaining the optimal solution according to the mode and median were the benchmark LA40,TA03, TA05 to TA08, and TA10. LA40 was the most complicated of these benchmarks, the optimalsolution was obtained in only one of the 30 executions, where the median and mode were furthestfrom the optimal value. Table 5 shows, based on the standard deviation that the data distribution ineach benchmark is completely homogeneous and the data are grouped very close to each other or withthe same optimal value. LA38 presents the data a little more dispersed (σ = 20.6).

Table 5. SACT results, medium benchmarks JSSP (15 × 15).

Problem OptimumUnits of Time

BetterUnits of Time

WorseUnits of Time


LA36 1268 1268 1281 1275 0 6.1 371.6 1278 1268LA37 1397 1397 1399 1397 0 0.9 230 1397 1397LA38 1196 1196 1245 1216 0 20.6 55 1218 1196LA39 1233 1233 1237 1234 0 1.8 54 1233 1233LA40 1222 1222 1234 1227 0 3.4 2245 1228 1229TA01 1231 1231 1231 1231 0 0.0 328 1231 1231TA02 1244 1244 1244 1244 0 0.0 501 1244 1244TA03 1218 1218 1223 1221 0 2.3 4373 1221 1223TA04 1175 1175 1175 1175 0 0.0 301 1175 1175TA05 1224 1224 1231 1229 0 2.8 2218 1230 1230TA06 1238 1238 1240 1239 0 0.8 5090 1239 1239TA07 1227 1227 1228 1228 0 0.4 99 1228 1228TA08 1217 1217 1224 1220 0 3.0 1986 1218 1218TA09 1274 1274 1281 1277 0 3.6 1433 1274 1274TA10 1241 1241 1253 1245 0 4.6 3830 1244 1244

Table 6 presents the MS results with SACT for large benchmarks. Most of the obtained values areclose to the upper bound (UB), four of the benchmarks achieve this value. TA28 finds the greatestrelative error (RE = 0.87%). TA22, TA30, YN3, and YN4 find the lowest RE = 0. The DMU benchmarks(46–50) are an exception because DMU49 finds the largest RE = 3.06 and DMU47 finds the lowest RE= 2.08. The largest standard deviation was 41.8 for the problem DMU08, and the lowest standarddeviation was 4.9 for the problem YN1; however, this value is less than half of the arithmetic mean(mean) and this represents little dispersion of the data, however the dispersion of the data found bySACT in a frequency distribution for large-sized benchmarks, is greater than the dispersion foundfor small-sized benchmark data and medium-sized benchmark data. The mean is greater than themedian and mode for DMU06 to DMU08 and DMU10, but with very close values, indicating verylittle skewed frequency distributions (see Figure 14). With DMU09 the mean is between the medianand the mode indicating that the frequency distribution is close to a symmetrical distribution (seeFigure 14d). The mean is less than the median and mode for DMU46 to DMU47 and DMU49, butwith very close values, indicating very little skewed frequency distributions (see Figure 15). WithDMU48 the mean is between the median and the mode indicating that the frequency distribution isclose to a symmetrical distribution without taking the atypical value of 3988 (see Figure 15c). WithDMU50 the mean is greater than the median and the mode but with very close values, indicating thatthe frequency distribution is close to a symmetrical distribution without taking the atypical value of3842 (see Figure 15e). The shortest time for TA benchmarks was for TA22 and the longest time was forTA30. The time for DMU benchmarks with the smallest relative error was for DMU08 and the time

Appl. Sci. 2019, 9, 3360 28 of 40

with the largest relative error was for DMU49. The shortest time for YN benchmarks with the smallestrelative error was for YN3 and the longest time with the biggest relative error was for YN2.

Table 6. SACT results, large benchmarks JSSP (20 × 20).

Problem UBUnits of Time

BetterUnits of Time

WorseUnits of Time


TA21 1642 1646 1772 1683 0.24 18.1 1938 1681 1665TA22 1600 1600 1680 1637 0 13.1 3476 1636 1644TA23 1557 1560 1628 1600 0.19 15.5 1681 1598 1598TA24 1646 1651 1693 1681 0.3 14.4 397 1683 1670TA25 1595 1597 1669 1633 0.13 17.9 2208 1634 1649TA26 1643 1651 1716 1684 0.49 15.6 4547 1681 1680TA27 1680 1682 1712 1701 0.12 6.5 1736 1700.5 1698TA28 1603 1617 1639 1625 0.87 6.6 1374 1623 1622TA29 1625 1627 1642 1631 0.12 4.9 4876 1628.5 1627TA30 1584 1584 1618 1607 0 6.0 6429 1608.5 1606

DMU06 3244 3254 3381 3321 0.31 30.4 267 3319 3307DMU07 3046 3065 3223 3127 0.62 37.5 3192 3122.5 3118DMU08 3188 3192 3385 3255 0.13 41.8 1696 3253 3202DMU09 3092 3121 3231 3174 0.94 29.5 1912 3173.5 3228DMU10 2984 3001 3084 3042 0.57 23.1 246 3041 3032DMU46 4035 4133 4189 4171 2.43 13.9 3424 4172 4176DMU47 3942 4024 4094 4070 2.08 13.9 1244 4073.5 4074DMU48 3763 3856 3988 3907 2.47 20.9 858 3906 3908DMU49 3710 3822 3871 3851 3.02 21.3 8301 3906 3902DMU50 3729 3829 3907 3882 2.68 16.2 5281 3881.5 3881

YN1 884 885 905 896 0.11 4.5 1558 895.5 896YN2 904 906 930 917 0.22 6.4 3353 916 913YN3 892 892 915 903 0 5.9 2459 903.5 904YN4 968 968 990 978 0 5.8 3525 977.5 974

Figure 16 presents the average of 5 tests of the relative error obtained for 15 × 15 problems. Itcan be noted that the maximum RE occurs in LA38 and was not greater than 1.7. The most difficultproblem was LA40, however 5 trials show that the average RE was not greater than 0.5. For the other15 × 15 problems, the maximum average RE is in the range 0 ≤ RE ≤ 0.6. The problems for which itwas easiest to find the global optimum were TA01, TA02, and TA04. In all the tests, they had RE = 0,which is why they are not present in Figure 16.


15 × 15 problems, the maximum average RE is in the range 0 ≤ RE ≤ 0.6. The problems for which it was easiest to find the global optimum were TA01, TA02, and TA04. In all the tests, they had RE = 0, which is why they are not present in Figure 16.

Figure 16. Relative error, average of five tests, JSSP 15 × 15.

Figure 17 presents the average of 30 tests of the relative error obtained for the YN problems. It can be observed that the maximum RE occurs in YN2 and was not greater than 1.7; this problem had more difficulty approaching the upper bound of agreement in Table 6. The lowest average RE was in YN4, not greater than 1.2 with respect to the upper bound in Table 6.

Figure 17. Relative error, average of thirty tests, JSSP 20 × 20.

Figure 18 presents the average of 30 tests of the relative error obtained for the DMU (06–10) problems. It can be observed that the maximum RE occurs in DMU07 and was not greater than 2.7. The DMU10 reached lowest average RE < 2. DMU08 was easier to approach the upper bound of agreement in Table 6. For the other DMU problems, the maximum average RE is in the range 2.3 < RE < 2.7.


Appl. Sci. 2019, 9, 3360 29 of 40

Figure 17 presents the average of 30 tests of the relative error obtained for the YN problems. It canbe observed that the maximum RE occurs in YN2 and was not greater than 1.7; this problem had moredifficulty approaching the upper bound of agreement in Table 6. The lowest average RE was in YN4,not greater than 1.2 with respect to the upper bound in Table 6.


15 × 15 problems, the maximum average RE is in the range 0 ≤ RE ≤ 0.6. The problems for which it was easiest to find the global optimum were TA01, TA02, and TA04. In all the tests, they had RE = 0, which is why they are not present in Figure 16.


Figure 17 presents the average of 30 tests of the relative error obtained for the YN problems. It can be observed that the maximum RE occurs in YN2 and was not greater than 1.7; this problem had more difficulty approaching the upper bound of agreement in Table 6. The lowest average RE was in YN4, not greater than 1.2 with respect to the upper bound in Table 6.


Figure 18 presents the average of 30 tests of the relative error obtained for the DMU (06–10) problems. It can be observed that the maximum RE occurs in DMU07 and was not greater than 2.7. The DMU10 reached lowest average RE < 2. DMU08 was easier to approach the upper bound of agreement in Table 6. For the other DMU problems, the maximum average RE is in the range 2.3 < RE < 2.7.


Figure 18 presents the average of 30 tests of the relative error obtained for the DMU (06–10)problems. It can be observed that the maximum RE occurs in DMU07 and was not greater than 2.7. TheDMU10 reached lowest average RE < 2. DMU08 was easier to approach the upper bound of agreementin Table 6. For the other DMU problems, the maximum average RE is in the range 2.3 < RE < 2.7.Appl. Sci. 2019, 9, x FOR PEER REVIEW 30 of 41

Figure 18. Relative error for thirty executions of problems DMU (06–10) of JSSP 20 × 20.

4.4. Behaviour in DMU Benchmarks Scheduling

Figure 19, presents the Gantt chart of the problem DMU08. The scheduling of the jobs in each of the problem’s machines can be observed. This type of distribution in the scheduling is characteristic of all problems, TA (21–30), YN (1–4), and DMU (06–10). The DMU problems (49–50) show different behavior. The scheduling distribution type is different. This is shown in Figure 20, where the Gantt diagram of the DMU46 problem is presented as an example.

Figure 19. Scheduling Gantt chart of the problem DMU08.


Appl. Sci. 2019, 9, 3360 30 of 40


Figure 19, presents the Gantt chart of the problem DMU08. The scheduling of the jobs in each ofthe problem’s machines can be observed. This type of distribution in the scheduling is characteristicof all problems, TA (21–30), YN (1–4), and DMU (06–10). The DMU problems (49–50) show differentbehavior. The scheduling distribution type is different. This is shown in Figure 20, where the Ganttdiagram of the DMU46 problem is presented as an example.




Figure 19, presents the Gantt chart of the problem DMU08. The scheduling of the jobs in each of the problem’s machines can be observed. This type of distribution in the scheduling is characteristic of all problems, TA (21–30), YN (1–4), and DMU (06–10). The DMU problems (49–50) show different behavior. The scheduling distribution type is different. This is shown in Figure 20, where the Gantt diagram of the DMU46 problem is presented as an example.

Figure 19. Scheduling Gantt chart of the problem DMU08. Figure 19. Scheduling Gantt chart of the problem DMU08.Appl. Sci. 2019, 9, x FOR PEER REVIEW 31 of 41

Figure 20. Scheduling Gantt chart of the problem DMU46

Figure 20 shows that the planning of the job sequences in DMU46 indicates that in machines 1 to 10, the work operations can be sequenced from time zero. For machines 11 to 20, the operations of jobs cannot be sequenced at time zero because there is a precedence order in operations of the same job. This precedence order makes it impossible to use machines 11 to 20 if the operations of machines 1 to 10 have not previously been executed.

4.5. Comparision of SACT with Other Algorithms

Table 7, presents the results for the FT (Fisher and Thompson) and LA (Lawrence) benchmarks. SACT and BRK-GA (Biased Random-Key Genetic Algorithm) algorithms find the best results reported in the literature. Both algorithms obtained the optimal value for LA40. With regard to the execution time, it shows that SACT is competitive for most benchmarks. Table A2, in Appendix A, shows the software/hardware environment used for the execution of the algorithms presented in Table 7.

Table 7. Efficiency and efficacy results for FT and LA benchmarks.

Prob Op ST t Sec BG t Sec SG t Sec AM t Sec TA t Sec SGS

t Sec

TGA t

Sec TS/PR

t Sec

UP t Sec HO t

Sec GT AG

%RE Size 10 × 10

FT10 930 0 1.05 0 10.1 -- -- 0 64.6 0 3.8 1.8 557 0 0.06 0 4.75 0 1208 0 4.1 0.54 0 LA16 945 0 5.1 0 4.6 0 38.8 -- -- -- -- 0 304 0 0.094 0 0.15 0 1458 0 19.9 0.11 0.10 LA17 784 0 4.9 0 4.6 -- -- -- -- -- -- -- -- 0 0.016 0 0.08 0 78 0 19.9 0 0 LA18 848 0 4.2 0 4.6 -- -- -- -- -- -- -- -- 0 0.015 0 0.09 0 76 0 19.9 0 0 LA19 842 0 4.1 0 4.6 0 34.6 -- -- 0 0.5 -- -- 0 0.025 0 0.16 0 1130 0 19.9 0 1.18 LA20 902 0 4.34 0 4.6 -- -- -- -- -- -- -- -- 0 0.031 0 0.11 0 1304 0 19.9 0.55 0.55

Size 15 × 15 LA36 1268 0 371.6 0 21.4 0 4655 0 36.6 0 9.9 -- -- 0 0.57 0 4.5 0.79 48,387 0 105 3.16 -- LA37 1397 0 230 0 21.4 0.29 4144 0 879.6 0 42.1 -- -- 0 0.51 0 26.2 0.72 49,836 0 105 6.59 -- LA38 1196 0 55 0 21.4 0.42 5049 0 55.4 0 47.8 -- -- 0 1.25 0 32.6 1.59 50,876 0 105 6.61 -- LA39 1233 0 54 0 21.4 -- -- 0 65.7 0 28.6 -- -- 0 0.5 0 11.6 1.38 50,603 0 105 4.62 -- LA40 1222 0 2245 0 21.4 0.33 4544 0.16 941.4 0.16 52.1 -- -- 0.16 0.86 0 385 0.57 50,609 0.16 105 2.46 --

Table 8, presents results for ORB, ABZ, TA, DMU, and YN benchmarks in [33]. SACT obtains the best results reported in the literature for most of the benchmarks. The relative error (RE) for ORB, ABZ, TA, YN, and DMU (06–10) benchmark is RE < 0.95%. For DMU (47–50,) benchmark is RE < 3.03%. This shows that SACT is competitive with respect to the other algorithms presented. In [36] the researchers report a new UB = 904 for the problem YN2; previously the known UB was 907. SACT finds a UB = 906. Although this is not the new value reported by [36], it was able to improve the UB

Figure 20. Scheduling Gantt chart of the problem DMU46

Figure 20 shows that the planning of the job sequences in DMU46 indicates that in machines 1 to10, the work operations can be sequenced from time zero. For machines 11 to 20, the operations of jobscannot be sequenced at time zero because there is a precedence order in operations of the same job.This precedence order makes it impossible to use machines 11 to 20 if the operations of machines 1 to10 have not previously been executed.

Appl. Sci. 2019, 9, 3360 31 of 40

4.5. Comparision of SACT with Other Algorithms

Table 7, presents the results for the FT (Fisher and Thompson) and LA (Lawrence) benchmarks.SACT and BRK-GA (Biased Random-Key Genetic Algorithm) algorithms find the best results reportedin the literature. Both algorithms obtained the optimal value for LA40. With regard to the executiontime, it shows that SACT is competitive for most benchmarks. Table A2, in Appendix A, shows thesoftware/hardware environment used for the execution of the algorithms presented in Table 7.

Table 7. Efficiency and efficacy results for FT and LA benchmarks.

Prob Op ST t Sec BG tSec SG t

Sec AM t Sec TA tSec SGS t

Sec TGA tSec TS/PR t

Sec UP t Sec HO tSec GT AG

%RE

Size 10 × 10

FT10 930 0 1.05 0 10.1 – – 0 64.6 0 3.8 1.8 557 0 0.06 0 4.75 0 1208 0 4.1 0.54 0LA16 945 0 5.1 0 4.6 0 38.8 – – – – 0 304 0 0.094 0 0.15 0 1458 0 19.9 0.11 0.10LA17 784 0 4.9 0 4.6 – – – – – – – – 0 0.016 0 0.08 0 78 0 19.9 0 0LA18 848 0 4.2 0 4.6 – – – – – – – – 0 0.015 0 0.09 0 76 0 19.9 0 0LA19 842 0 4.1 0 4.6 0 34.6 – – 0 0.5 – – 0 0.025 0 0.16 0 1130 0 19.9 0 1.18LA20 902 0 4.34 0 4.6 – – – – – – – – 0 0.031 0 0.11 0 1304 0 19.9 0.55 0.55

Size 15 × 15

LA36 1268 0 371.6 0 21.4 0 4655 0 36.6 0 9.9 – – 0 0.57 0 4.5 0.79 48,387 0 105 3.16 –LA37 1397 0 230 0 21.4 0.29 4144 0 879.6 0 42.1 – – 0 0.51 0 26.2 0.72 49,836 0 105 6.59 –LA38 1196 0 55 0 21.4 0.42 5049 0 55.4 0 47.8 – – 0 1.25 0 32.6 1.59 50,876 0 105 6.61 –LA39 1233 0 54 0 21.4 – – 0 65.7 0 28.6 – – 0 0.5 0 11.6 1.38 50,603 0 105 4.62 –LA40 1222 0 2245 0 21.4 0.33 4544 0.16 941.4 0.16 52.1 – – 0.16 0.86 0 385 0.57 50,609 0.16 105 2.46 –

Table 8, presents results for ORB, ABZ, TA, DMU, and YN benchmarks in [33]. SACT obtains thebest results reported in the literature for most of the benchmarks. The relative error (RE) for ORB, ABZ,TA, YN, and DMU (06–10) benchmark is RE < 0.95%. For DMU (47–50,) benchmark is RE < 3.03%. Thisshows that SACT is competitive with respect to the other algorithms presented. In [36] the researchersreport a new UB = 904 for the problem YN2; previously the known UB was 907. SACT finds a UB = 906.Although this is not the new value reported by [36], it was able to improve the UB known before2014. In efficiency, SACT is competitive with the ACOFT-MWR (Ant Colony Optimization with FastTaboo - Most Work Remaining) . The exception can be seen with BRK-GA algorithm for the DMUbenchmarks, SACT is, 0.12 < RE < 3.04 and BRK-GA is, 0 ≤ RE < 0.49. Table A2, in Appendix A, showsthe software/hardware environment used for the execution of the algorithms presented in Table 8.

Appl. Sci. 2019, 9, 3360 32 of 40

Table 8. Efficiency and efficacy for orb, ABZ, TA, DMU, and YN benchmarks.

Problem Op/UBST t Sec BG t Sec AM t Sec TSSA t Sec SGS T Sec TGA t Sec IO t Sec TS/PR t Sec UP t Sec GT AG

%RE

Size 10 × 10

ORB01 1059 0 4.34 0 5.8 0 56.6 0 3.5 0 342 0 0.06 – – 0 0.51 0 2312 2.36 3.12ORB02 888 0 9720 0 5.8 0 569.3 0 6.4 0.1 306 0 0.06 – – 0 1.69 0.11 2393 0.23 0.68ORB03 1005 0 1048 0 5.8 0 403.7 0 13.8 1.2 330 0 0.15 – – 0 1.46 0 2358 3.18 2.39ORB04 1005 0 155 0 5.8 0 17.9 0 14.3 0 306 0 0.45 – – 0 3.71 0 796 2.29 1.09ORB05 887 0 1354 0 5.8 0 670.3 0 6.6 0 366 0 0.76 – – 0 7.28 0.23 2458 0.79 1.58ORB06 1010 0 59 0 5.8 – – 0 8.5 – – 0 0.72 – – 0 1.81 0.30 2525 2.48 1.78ORB07 397 0 4.95 0 5.8 – – 0 0.5 – – 0 0.02 – – 0 0.13 0 2096 1.76 2.02ORB08 899 0 10.33 0 5.8 – — 0 7.2 – – 0 0.09 – – 0 3.99 0 2338 4.23 1.67ORB09 934 0 5 0 5.8 – – 0 0.4 – – 0 0.09 – – 0 0.47 0 884 0.96 0.96ORB10 944 0 4.84 0 5.8 – – 0 0.3 – – 0 0.03 – – 0 0.09 0 817 2.44 –ABZ5 1234 0 12.62 – – 0 501.9 – – – – 0 0.04 – – – – – – 0.32 –ABZ6 943 0 4.23 – – 0 199.3 – – – – 0 0.03 – – – – – – 0.42 –

Size 15 × 15

TA01 1231 0 328 0 30.4 0 1531.4 0 11.2 3.1 2782 – – 0 124 0 2.93 – – – –TA02 1244 0 501 0 30.4 0 685.2 0 30.1 – – – – 0 118 0 38 – – – –TA03 1218 0 4373 0 30.4 0.16 1833.7 0 108.5 – – – – 0 120 0 44 – – – –TA04 1175 0 301 0 30.4 0 1186.2 0 71.7 – – – – 0 117 0 39 – – – –TA05 1224 0 2218 0 30.4 0.33 1492.6 0 10.8 – – – – 0 120 0 11 – – – –TA06 1238 0 5090 0 30.4 0 1549.1 0 125.2 – – – – 0 113 0 178 – – – –TA07 1227 0.08 99 0.081 30.4 0.08 1687 0.08 138.6 – – – – 0 117 0.08 0.60 – – – –TA08 1217 0 1986 0 30.4 0 968.4 0 27.6 – – – – 0 108 0 2.43 – – – –TA09 1274 0 1433 0 30.4 0 1694.2 0 61.3 – – – – 0 127 0 19 – – – –TA10 1241 0 3380 0 30.4 0 1418.2 0 68 – – – – 0 122 0 42 – – – –

Size 20 × 20

TA21 1642 0.24 1938 0 143.2 0.31 4158.4 0.12 437 – – – – 0 408 0.12 503 – – – –TA22 1600 0 3476 0 143.2 0.06 3586.4 0 433.5 – – – – 0 395 0 229 – – – –TA23 1557 0.19 1681 0 143.2 0.19 4175.7 0.19 429.4 – – – – 0 390 0 360 – – – –

Appl. Sci. 2019, 9, 3360 33 of 40

Table 8. Cont.

Problem Op/UBST t Sec BG t Sec AM t Sec TSSA t Sec SGS T Sec TGA t Sec IO t Sec TS/PR t Sec UP t Sec GT AG

%RE

TA24 1644 0.43 397 0.12 143.2 0.49 3320.2 0.12 431.6 – – – – 0.12 435 0.06 779 – – – –TA25 1595 0.13 2208 0 143.2 0.13 3654.3 0.13 421 – – – – 0 414 0 416 – – – –TA26 1643 0.49 4547 0 143.2 0.55 3178.8 0.24 436.2 – – – – 0 87 0.24 268 – – – –TA27 1680 0.12 1736 0 143.2 0.36 3523.8 0 447.8 – – – – 0 423 0 255 – – – –TA28 1603 0.87 1374 0 143.2 0.94 3804.8 0 431.2 – – – – 0 370 0.62 326 – – – –TA29 1625 0.12 4876 0 143.2 0.12 3324.9 0.12 426.2 – – – – 0 396 0 94 – – – –TA30 1584 0 6429 0 143.2 0.69 4003.5 0 436.1 – – – – 0 429 0 389 – – – –

DMU06 3244 0.31 267 0 145.4 – – – – – – – – – – 0.03 823 – – – –DMU07 3046 0.62 3192 0 145.4 – – – – – – – – – – 0 361 – – – –DMU08 3188 0.13 1696 0 145.4 – – – – – – – – – – 0 296 – – – –DMU09 3092 0.94 1912 0 145.4 – – – – – – – – – – 0.07 148 – – – –DMU10 2984 0.57 246 0 145.4 – – – – – – – – – – 0.03 253 – – – –DMU46 4035 2.43 3424 0 187.7 – – – – – – – – – — 0 985 – – – –DMU47 3939 2.15 1244 0 187.7 – — – – – – – – – – 0.08 829 – – – –DMU48 3763 2.47 858 0.48 187.7 – - – – – – – – – – 0.40 939 – – – –DMU49 3710 3.02 8301 0.35 187.7 – – – – – – – – – – 0 634 – – – –DMU50 3729 2.68 5281 0.08 187.7 – – – – – – – – – – 0 610 – – – –

YN1 884 0.11 1558 0 105.2 – – 0 106.6 0.2 15,786 0.23 92.8 0 190 0 169 – – –YN2 904 0.22 3353 0 105.2 – – 0.33 110.4 4.4 14,586 0.77 13.1 0 197 0 202 – – – –YN3 892 0 2459 0 105.2 – – 0 110.8 1.3 16,662 0.56 37.2 0 212 0 344 – – – –YN4 967 0.1 3525 0.1 105.2 – – 0.21 108.7 2.17 14,752 0.83 114.1 0.10 — 0.10 321 – – – –

Appl. Sci. 2019, 9, 3360 34 of 40

Table 9, presents the relative error of algorithms with threads/processes for the FT, LA, ORB,TA, DMU, and YN benchmarks. SACT has better performance with respect to the other algorithms,except DMU09. SACT has a relative error in the range of 0 ≤ RE ≤ 0.94%, PPSO (Parallel ParticleSwarm Optimization) has a relative error in the range of 29 ≤ RE ≤ 68%, CGA-PR (Coarse-GrainedGenetic Algorithm with Path-Relinking) has a relative error in the range of 0 ≤ RE ≤ 2.49%, PaGA(Parallel Agent-Based Genetic Algorithm) has a relative error in the range of 0 ≤ RE ≤ 12.4%, HGAPSA(Hybridization of Genetic Algorithm with Parallel Implementation of Simulated Annealing) presentsa relative error of 0.79 ≤ RE ≤ 1.92%, HIMGA presents 0 ≤ RE ≤ 1.01%, NIMGA presents 0 ≤ RE ≤3.01%, IIMMA presents 0 ≤ RE ≤ 0.55% and AGS presents 0 ≤ RE ≤ 1.76%. Concerning the HGACCdistributed algorithm, SACT offers a better performance for YN benchmarks. It reaches the sameeffectiveness for 10 × 10 and 15 × 15 size benchmarks. Thus, it is best for the DMU06 benchmark and iscompetitive for the other DMU benchmarks. Table A2, in Appendix A, shows the software/hardwareenvironment used for the execution of the algorithms presented in Table 9.

Table 9. Parallel algorithms, efficacy for FT, LA, ORB, TA, DMU, and YN benchmarks.

Problem Size Op/UB%RE

SACT PPSO cGA-PR PaGA HGAPSA HIMGA NIMGA IIMMA PGS PABC HG

FT06 6 × 6 55 0 – – 0 – 0 0 0 0 0 –FT10 10 × 10 930 0 – 0 7.2 – 0 0 0 0.9 0 0LA16 10 × 10 945 0 29 – 5.2 – 0 0.11 0 – 0 0LA17 10 × 10 784 0 33 – 1.2 – 0 0 0 – 0 0LA18 10 × 10 848 0 33 – 1.4 – 0 0 0 – 0 0LA19 10 × 10 842 0 37 – 3.7 – 0 0 0 – 0 0LA20 10 × 10 902 0 32 – 1.1 – 0 0.55 0 – 0.55 0LA36 15 × 15 1268 0 65 – – 0.87 0 1.97 0 – – 0LA37 15 × 15 1397 0 58 – – 0.79 0 3.01 0 – – 0LA38 15 × 15 1196 0 68 1.0 – 1.92 0 2.17 0 – – 0LA39 15 × 15 1233 0 67 – – 1.05 0 2.11 0 – – 0LA40 15 × 15 1222 0 68 1.31 – 1.56 0.16 1.96 0.16 – – 0

ORB01 10 × 10 1059 0 – 0 8.5 – 0 0 0 0 – 0ORB02 10 × 10 888 0 – – 4.6 – 0 0.23 0 0.1 – 0ORB03 10 × 10 1005 0 – 0 12.3 – 0 2.09 0 0 – 0ORB04 10 × 10 1005 0 – 0 5.7 – 0 1.39 0 0 – 0ORB05 10 × 10 887 0 – – 5.5 – 0 0.68 0 0 – 0ORB06 10 × 10 1010 0 – – 4.0 – 0 0.20 0 – – 0ORB07 10 × 10 397 0 – – 4.8 – 0 0 0 – – 0ORB08 10 × 10 899 0 – 0 12.4 – 0 1.11 0 – – 0ORB09 10 × 10 934 0 – – 6.4 – 0 0.86 0 – – 0ORB10 10 × 10 944 0 – 0 – – 0 – 0 – – 0TA21 20 × 20 1642 0.24 – 0.49 – – 0.49 – – – – –TA22 20 × 20 1600 0 – 0.38 – – – – – – – –TA23 20 × 20 1557 0.19 – 0.19 – – – – – – – –TA24 20 × 20 1646 0.3 – 0.36 – – – – – – – –TA25 20 × 20 1595 0.13 – 0.13 – – – – – – – –TA26 20 × 20 1643 0.49 – 0.55 – – – – – – – –TA27 20 × 20 1680 0.12 – 0.36 – – – – – – – –TA28 20 × 20 1603 0.87 – 0.87 – – – – – – – –TA29 20 × 20 1625 0.12 – 0.25 – – – – – – – –TA30 20 × 20 1584 0 – 0 – – – – – – – –

DMU06 20 × 20 3244 0.31 – 0.52 – – – – – – – 0.74DMU07 20 × 20 3046 0.62 – 1.15 – – – – – – – 0.59DMU08 20 × 20 3188 0.13 – 0.53 – – – – – – – 0DMU09 20 × 20 3092 0.94 – 0.13 – – – – – – – 0.59DMU10 20 × 20 2984 0.57 – 0.84 – – – – – – – 0.10

YN1 20 × 20 884 0.11 – 2.49 – – 1.01 – 0.22 1.4 – 0.23YN2 20 × 20 904 0.22 – 1.77 – – 0.99 – 0.55 1.2 – 0.33YN3 20 × 20 892 0 – 1.01 – – 0.90 – 0.34 0.9 – 0YN4 20 × 20 967 0.1 – 1.45 – – 0.93 – 0.21 1.76 – 0.21

4.6. SACT Computational Efficiency

Figure 21 shows the SACT speedup for the YN benchmarks. The communication throughcooperation between threads (access to the critical section) affects the speedup. An increased numberof threads lead to greater cooperation because each running thread has the need to deposit the best

Appl. Sci. 2019, 9, 3360 35 of 40

solution in a critical section shared by all threads (see Figure 3), generating a bottleneck where in aninstant of time the threads will have to wait their turn to access the critical section. This implies thatincreasing the number of threads will have a higher bottleneck, which results that the speedup movesaway from the ideal. The speedup for YN2 is farthest from ideal with respect to other YN benchmarks.This may be because YN2 is taken more time to generate feasible solutions with the neighborhoodgeneration mechanism [31]. So, it is possible that YN1, YN3, and YN4 will find solutions feasiblefaster. The neighborhood mechanism used in SACT is the only place in the algorithm that can performdifferently on each benchmark. The speedup is very good for all YN benchmarks since is close tothe ideal.


4.6. SACT Computational Efficiency

Figure 21 shows the SACT speedup for the YN benchmarks. The communication through cooperation between threads (access to the critical section) affects the speedup. An increased number of threads lead to greater cooperation because each running thread has the need to deposit the best solution in a critical section shared by all threads (see Figure 3), generating a bottleneck where in an instant of time the threads will have to wait their turn to access the critical section. This implies that increasing the number of threads will have a higher bottleneck, which results that the speedup moves away from the ideal. The speedup for YN2 is farthest from ideal with respect to other YN benchmarks. This may be because YN2 is taken more time to generate feasible solutions with the neighborhood generation mechanism [31]. So, it is possible that YN1, YN3, and YN4 will find solutions feasible faster. The neighborhood mechanism used in SACT is the only place in the algorithm that can perform differently on each benchmark. The speedup is very good for all YN benchmarks since is close to the ideal.

Figure 21. SACT algorithm, speedup for benchmarks 20 × 20 (YN1, YN2, YN3, and YN4).

5. Conclusions

The study presented evidence that a benefit exists when the cooperation of threads is used applying effective-address procedure in the algorithm of simulated annealing for the job shop scheduling problem. It is also observed that the SACT algorithm can work more efficiently if it is executed in a computer with a greater number of threads per core. This permits an increase in the number of effective-address upon executing a greater number of SA in parallel.

Working with SACT in parallel allows the effective-address procedure to be efficiently applied because each thread constantly updates its search direction to a good solution space. In addition, no thread waits for any other thread in its simulated annealing run. That is why the search for better solutions between threads is asynchronous, enabling to run a larger number of simulated annealing compared to other threads, but always sharing the best solution space, and trying to improve the search direction with effective-address procedure on a constant basis. This makes the SACT algorithm more efficient if the number of threads is greater, which is reflected in the experimental results, as shown in Section 4.2.

According to the experimental results, it can be concluded that by applying cooperation threads with a directed path toward the best solution, a larger number of threads find best solutions. Since

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

IdealYN1

YN2

Number of threads

Spee

dUp

Figure 21. SACT algorithm, speedup for benchmarks 20 × 20 (YN1, YN2, YN3, and YN4).

5. Conclusions

The study presented evidence that a benefit exists when the cooperation of threads is usedapplying effective-address procedure in the algorithm of simulated annealing for the job shop schedulingproblem. It is also observed that the SACT algorithm can work more efficiently if it is executed ina computer with a greater number of threads per core. This permits an increase in the number ofeffective-address upon executing a greater number of SA in parallel.

Working with SACT in parallel allows the effective-address procedure to be efficiently appliedbecause each thread constantly updates its search direction to a good solution space. In addition, nothread waits for any other thread in its simulated annealing run. That is why the search for bettersolutions between threads is asynchronous, enabling to run a larger number of simulated annealingcompared to other threads, but always sharing the best solution space, and trying to improve thesearch direction with effective-address procedure on a constant basis. This makes the SACT algorithmmore efficient if the number of threads is greater, which is reflected in the experimental results, asshown in Section 4.2.

According to the experimental results, it can be concluded that by applying cooperation threadswith a directed path toward the best solution, a larger number of threads find best solutions. Since thediversity of solutions increases as the number of threads increases, a quick convergence before findinggood solutions is avoided.

Appl. Sci. 2019, 9, 3360 36 of 40

When the cooperation thread is applied without a directed path towards the same solution, findinga good result does not depend on the number of threads running on SACT. The value of the makespanis very similar for any number of threads.

The statistical review for the 19 small-sized benchmarks shows that SACT is highly effective sincethe optimal value is found in all instances. In 16 instances, the mean is the optimum, which indicatesthat for these instances, each of the 30 tests performed by SACT obtained the optimal value. Accordingto the result of the median, it is understood that in 18 instances, the optimal solution is found in at leasthalf of the 30 tests performed for each, which is reaffirmed by the mode of the same 18 instances.

The statistical review for the 15 medium-sized benchmarks shows that SACT is highly effectivesince the optimal value is found in all instances. In 4 instances, the mean is the optimum and thisindicates that for these instances, each of the 30 tests performed by SACT obtained the optimal value.According to the result of the median, it is understood that in 5 instances, the optimal solution is foundin at least half of the 30 tests performed for each. In 8 instances, the mode indicates that the optimalvalue is the one that appears more frequently.

The statistical review for the 20 large-sized benchmarks shows that SACT is very effective. In 4instances, the upper bound is found and in 15 instances, the relative error does not exceed 1%.

Because the execution in SACT is executed by threads and communication in the algorithmhandles only two critical sections, the efficiency (speedup) of the proposed algorithm is close to theideal value. SACT also shows that efficacy is competitive when compared with other algorithms in theliterature that have been very successful. Additionally, it is better than the algorithms in the literaturethat use threads/processes for the job shop problem.

Finally, the conclusion can be drawn that the optimum number of threads to execute in parallel bySACT is a parameter that must be tuned due to this cooperation in the threads.

The planning characteristics of the DMU (46–50) problems increase the degree of difficulty inobtaining good solutions by the proposed SACT algorithm. This is a topic of interest for futureresearch, to search for an improved neighborhood mechanism that finds better solutions with theSACT algorithm for problems that behave like the DMU (46–50) problems with respect to the operationprecedence and restrictions that prohibit starting all operations at time zero.

Author Contributions: Conceptualization, M.A.C.-C.; Formal analysis, J.d.C.P.-A.; Investigation, M.A.C.-C. andM.H.C.-R.; Methodology, M.A.C.-C. and J.d.C.P.-A.; Software, M.A.C.-C. and M.H.C.-R.; Validation, J.d.C.P.-A.and M.H.C.-R.; Writing – review & editing, M.A.C.-C.

Funding: This research was funded by PRODEP grant number SA-DDI-UAEM/15/451.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Table A1. Benchmarks used to evaluate the proposed SACT algorithm.

ProblemSize Optimum/UB

Units of TimeJobs Machines

FT6 6 6 55

FT10 10 10 930

LA16 10 10 945

LA17 10 10 784

LA18 10 10 848

LA19 10 10 842

LA20 10 10 902

Appl. Sci. 2019, 9, 3360 37 of 40

Table A1. Cont.



ORB01 10 10 1059

ORB02 10 10 888

ORB03 10 10 1005

ORB04 10 10 1005

ORB05 10 10 887

ORB06 10 10 1010

ORB07 10 10 397

ORB08 10 10 899

ORB09 10 10 934

ORB10 10 10 944

ABZ5 10 10 1234

ABZ6 10 10 943

LA36 15 15 1268

LA37 15 15 1397

LA38 15 15 1196

LA39 15 15 1233

LA40 15 15 1222

TA01 15 15 1231

TA02 15 15 1244

TA03 15 15 1218

TA04 15 15 1175

TA05 15 15 1224

TA06 15 15 1238

TA07 15 15 1227

TA08 15 15 1217

TA09 15 15 1274

TA10 15 15 1241

TA21 20 20 1642

TA22 20 20 1600

TA23 20 20 1557

TA24 20 20 1646

TA25 20 20 1595

TA26 20 20 1643

TA27 20 20 1680

TA28 20 20 1603

TA29 20 20 1625

TA30 20 20 1584

DMU06 20 20 3244

DMU07 20 20 3046

DMU08 20 20 3188

DMU09 20 20 3092

DMU10 20 20 2984

DMU46 20 20 4035

DMU47 20 20 3942

DMU48 20 20 3763

DMU49 20 20 3710

Appl. Sci. 2019, 9, 3360 38 of 40

Table A1. Cont.



DMU50 20 20 3729

YN1 20 20 884

YN2 20 20 904

YN3 20 20 892

YN4 20 20 968

Table A2. Software/hardware environment used in algorithms with which SACT is compared.

Algorithm Hardware and Software

PPSO, [5] Server and client Machines, Logical ring topology, Java, Windows system

HGAPSA, [14] Server and client Machines

cGA-PR, [15] Workstation Pentium IV, multicore, 2.0GHz, 1GB, Microsoft Visual C++

PaGA, [19] Computer network with JADE Middleware, Java

HIMGA, [20] PC, 3.4GHz, Intel®, Core(TM), i7-3770 CPU, 8GB, C++

NIMGA. PC, [21] PC, 3.4GHz, Intel®, Core(TM), i7-3770 CPU, 8GB, C++

IIMMA, PC, [22] PC, 3.4 GHz, Intel®, Core(TM), i7-3770 CPU, 8GB, C++

Sequential AntGenSA (SGS),Parallel AntGenSA (PGS), [23] Cluster 4nodes, Intel® Xeon® 2.3 GHz, 64GB, Linux CentOS, C, OpenMP

PABC, [24] Four computers system configuration, JAVA

HGACC (HG), [25] CLUSTER, 48 cores, Xeon 3.06GHz, Linux Centos 5.5, GNU gcc, MPI Library

BRK-GA (BG), [36] AMD Opteron 2.2GHz CPU, Linux Fedora release 12, C++

SAGen (SG), [37] Pentium 120 (0.12 GHz), Pentium 166

ACOFT-MWR (AM), [38] PC AMD 1533MHz CPU, 768 MB, Windows XP, Microsoft Visual C++ 6.0

TSSA (TA), [39] PC Pentium IV 3.0GHz, Visual C++

HPSO (HO), [40] PC, AMD Athlon 1700+ (1.47 GHz), Visual C++

TGA, [41] PC 2.2 GHz, 8GB RAM, GNU gcc compiler

IEBO (IO), [42] 2.93 GHz, Intel Xeon X5670, GNU g++ compiler

TS/PR, [43] PC Quad-Core AMD Athlon 3 GHz, 2GB, Windows 7, C++

UPLA, [44] (UP) Intel CoreTM i5, processor M580 2.67 GHz, 6GB, C#

ALSGA (AG), [45] Intel core 2 duo, 2.93 GHz, 2.0GB, Java Agent DEvelopment platform (JADE)

GA-CPG-GT (GT), [46] PC 3.40 GHz Intel(R) Core (TM) i7-3770, 8GB, C++

SACT (ST), this workWorkstation PowerEdge T320, Intel® Xeon® Processor E5-2470 v2, 10cores,3.10 GHz each, 24GB, Windows Vista Ultimate 64 bits O.S, Visual C++ 2008,

MFC library

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